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+{"text":"\\section{Introduction}\n\nThe dynamics of a quantum system composed by $N$ particles in $\\bR^d$, $d=1,2,3$, interacting via a zero-range, two-body interaction is described by the formal Hamiltonian\n\\beq\\label{hamnd}\n\\mathcal H =- \\sum_{i=1}^N \\f{1}{2m_i} \\Delta_{\\xv_i} + \\sum_{\\underset{i < j}{i,j=1}}^N \\mu_{ij} \\,\\delta(\\xv_i - \\xv_j),\n\\eeq\nwhere $\\xv_i \\in \\bR^d$, $i=1,\\ldots,N$,  is the coordinate of the $i$-th particle,   $m_i$ is the corresponding  mass, $\\Delta_{\\xv_i}$ is the Laplacian relative to  $\\xv_i $, and $\\mu_{ij} \\in \\bR$ is the strength  of the interaction between particles $i$ and $j$. To simplify the notation we set $\\hbar =1$. \nFormal Hamiltonians of the type \\eqref{hamnd} are widely used in physical applications. In particular they are relevant in the study of ultra-cold quantum gases, both in the bosonic and in the fermionic case, in the so-called unitary limit, i.e., for infinite two-body scattering length (see \\cite{bh},\\cite{wc1},\\cite{wc2},\\cite{cmp} and references therein).\n\nThe first step towards a rigorous approach to the analysis of the model is to give \nthe  mathematical definition  of such a Hamiltonian as a self-adjoint operator on the appropriate $L^2$-space. One first notices that  the interaction term in \\eqref{hamnd} is effective  only on the hyperplanes $\\cup_{i<j} \\{\\xv_i = \\xv_j\\}$. This suggests to consider the operator $\\dot{\\mathcal H}_0$ defined as the free Hamiltonian restricted to a domain of smooth functions vanishing in the neighbourhood of each hyperplane $\\{\\xv_i=\\xv_j\\}$. It is easily seen that $\\dot{\\mathcal H}_0$ is symmetric but not self-adjoint and a trivial self-adjoint extension is the free Hamiltonian on its natural domain. Then, by definition,    any non trivial self-adjoint extension of the operator $\\dot{\\mathcal H}_0$ is  a  Hamiltonian for a system of $N$ quantum particles in $\\bR^d$ with a two-body, zero-range interaction. \n \nThe second and more relevant problem  is the concrete construction of such self-adjoint extensions. It turns out that each extension is characterized by a specific generalized boundary condition satisfied by the wave function at the hyperplanes. \nThe two most frequently used techniques for the construction are Krein's theory of self-adjoint extensions and the approximation by regularized Hamiltonians, in the sense of the limit of the resolvent or of the quadratic form. The difficulty of the analysis depends on the dimension $d$. For $d=1$ the problem is greatly simplified by the fact that the interaction term is a small perturbation of the free Hamiltonian in the sense of quadratic forms. For $d=2$ a natural class of Hamiltonians with local zero-range interactions was constructed in \\cite{DFT} exploiting renormalised quadratic forms and it was also shown that such Hamiltonians are all bounded from below. For $d=3$ the problem is more delicate. In order to illustrate the main point, let us consider the special  case $n=2$  where, in the center of mass  reference frame,  one is reduced to study a one-body problem in the relative coordinate $\\xv$  with a  fixed $\\delta$-interaction placed at the origin. In this case the problem is completely understood (see, e.g., \\cite{al}) and the entire class of  Hamiltonians \ncan be explicitly constructed. It turns out that the domain of each Hamiltonian  consists of functions  $ \\psi \\in L^2(\\bR^3) \\cap H^2(\\bR^3 \\setminus \\{0\\})$ which exhibit the following singular behaviour for $  |\\xv| \\rightarrow 0$\n\\beq\\label{bc0}\n\\psi(\\xv) = \\f{q}{|\\xv |} + r  + o(1)\\, ,\t\\hspace{1cm}\t\\text{with}\\;\\; \\;r=\\alpha\\, q\\, ,\n\\eeq\nwhere $q \\in \\ci$ and $\\alpha \\in \\bR$ is a parameter proportional to the inverse of the scattering length. We underline that the relation $r=\\alpha q$ in \\eqref{bc0} should be understood as the generalized boundary condition satisfied at the origin by all the elements of the domain. In the general case $N>2$ the  characterization of all possible self-adjoint extensions of $\\dot{\\mathcal H}_0$ is more involved. However,  a  class of extensions  based on the analogy with the case $N=2$ can be explicitly constructed. More precisely, one considers the so-called Skornyakov-Ter-Martirosyan (STM) extension of $\\dot{\\mathcal H}_0$ which, roughly speaking, is a symmetric operator acting on functions $\\psi \\in L^2(\\bR^{3N}) \\cap H^2( \\bR^{3N} \\setminus \\cup_{i<j} \\{ \\xv_i=\\xv_j\\})$\nsatisfying the following condition for $|\\xv_i - \\xv_j| \\rightarrow 0$:\n\\beq\\label{bcn}\n\\psi(\\xv_1, \\ldots , \\xv_n)= \\f{q_{ij}}{|\\xv_i - \\xv_j|} + r_{ij} + o(1)\\,,\t\\hspace{1cm} \\text{with} \\;\\;\\; r_{ij}= \\alpha_{ij} q_{ij}\\, ,\n\\eeq\nwhere $q_{ij}$ is a  suitable function defined on the hyperplane $\\{\\xv_i = \\xv_j\\}$ and $\\{\\alpha_{ij}\\} $ is  a collection of real parameters labelling the extension.  Noticeably, the boundary condition \\eqref{bcn} defining the STM extension of $\\dot{\\mathcal H}_0$ is a  natural  generalization to the case $N>2$ of the condition \\eqref{bc0} that characterizes  the two-body case.   Unfortunately, unlike \\eqref{bc0}, \\eqref{bcn} does not necessarily define a self-adjoint operator. Indeed, for a system of three identical bosons it was shown in \\cite{fm} that the STM extension is not self-adjoint and all  its self-adjoint extensions are unbounded from below owing to the presence of an infinite sequence of energy levels $E_k$ going to $-\\infty$ for $k \\rightarrow \\infty$. In \\cite{MM} this result was generalized  to the case of three distinguishable  particles with different masses. This kind of instability  is known in the literature as the Thomas effect.  \nIt should be stressed that the Thomas effect is strongly related to the well-known Efimov effect (see, e.g., \\cite{bh}, \\cite{adfgl}, \\cite{ahkw}) even if, to our knowledge, a rigorous mathematical investigation of this  connection is still lacking. We also mention that if, instead of \\eqref{bcn},   one introduces a ``non-local'' boundary condition on the hyperplanes then it is possible to construct a positive Hamiltonian and to study its stability properties for $N$ large (see, e.g., \\cite{fs}). In this paper we do not consider this kind of Hamiltonians.\n \nIt is reasonable to expect that the Thomas effect does not occur if the Hilbert space of states is suitably restricted, e.g., introducing  symmetry constraints on the wave function. A remarkably important constraint is antisymmetry. In fact, a wave function that is antisymmetric under exchange of coordinates of two particles necessarily vanishes at the coincidence points of such two particles, thus making their mutual zero-range interaction ineffective. Analogously, in a mixture of fermions of different species subject to pairwise zero-range interaction, fermions of the same species cannot ``feel'' mutual zero-range interaction and therefore the interaction term in the Hamiltonian is less singular.\n\n\nIn this paper we consider the simplified model consisting of  $N$ identical fermions, with unit mass, and a different particle with mass $m$, interacting with the fermions through a zero-range potential. For such a model only partial results are available and it is remarkable that they strongly depend on the parameters $N$ and $m$. \n\nConcerning the physical literature, we mention that for $N=2$ it is known (see, e.g., \\cite{bh} and references therein)  that for $m < 0.0735= (13.607)^{-1}$ the Thomas effect is present while for $m> 0.0735$ the STM extensions are expected to be bounded from below. More recently (see \\cite{cmp}), it was shown by means of analytical and numerical arguments that in the case $N=3$ the Thomas effect occurs for $m< 0.0747= (13.384)^{-1}$. This, in particular, indicates that for $0.0735<m< 0.0747$ there is a sequence of  genuine four-body bound states with energy going to $-\\infty$.\n\nFrom the rigorous point of view, the case $N=2$ was investigated first in \\cite{m1} and \\cite{MS}, where it was proved that the STM extension is self-adjoint if $m=1$. In \\cite{Sh}, the existence of a critical mass  $m^*(2) \\simeq 0.0735$ was shown such that for $m<m^*(2)$ any STM extension -- more precisely its restriction to the subspace of angular momentum $l=1$ -- is not self-adjoint and its self-adjoint extensions are unbounded from below. Therefore, the system is unstable and the Thomas effect occurs. This result was extended in \\cite{FT}, where  it was shown that the quadratic form associated with the STM extension, restricted to the subspace of angular momentum $l$, is unbounded from below if an explicit condition on $m$ and $l$ is satisfied. Finally, in the case $N\\leq 4$ and $m$ sufficiently large it was shown in \\cite{m3} that the STM extension is self-adjoint.\n\nIn the present work we study the problem for generic $N$ and $m$ and following the line of \\cite{DFT} we construct a renormalised quadratic form $\\mathcal F_{\\alpha}$, $\\alpha \\in \\bR$, which is naturally associated with the STM symmetric extension  $H_{\\alpha}$. Here $\\alpha$ is the value that all the parameters $\\alpha_{ij}$ labelling the STM extension must be equal to as a consequence of the fermionic symmetry. \nNote that in the bosonic case the quadratic form would differ in a sign in front the non-diagonal term (see the remark after \\eqref{oqform}).\n\nThe first question we address is a non-trivial sufficient condition for the \\emph{stability} of the model. As a first main result (Theorem \\ref{clbou}) we prove that for any $N$ there is  a value of the mass $m^*(N)>0$  such that $\\mathcal F_{\\alpha}$ is closed and bounded from below if $m>m^*(N)$. This implies that $\\mathcal F_{\\alpha}$ is the quadratic form of a \\emph{unique} self-adjoint extension of the STM operator $H_{\\alpha}$, and this extension is bounded from below. It therefore describes a stable system, where the Thomas effect does not occur.\n\nSuch a critical mass  was first conjectured in \\cite{m2}  and is precisely the unique root of an explicit equation (see \\eqref{Lambda} and \\eqref{La=1} in Section \\ref{main results: sec}). It turns out that $m^*(N)$ is increasing with $N$ and that the condition $m>m^*(N)$ guarantees the stability also in the limit of infinitely many fermions, provided that the mass of the extra particle scales as $m \\propto N$. \n\n\n\nThe second question we address is a sufficient condition for the \\emph{instability} of the model. This can be seen by plugging suitable trial functions into $\\mathcal F_{\\alpha}$. An attempt in this direction is in \\cite[Section 7]{DFT}, but with trial functions that do not satisfy the fermionic symmetry: thus, the result stated there on the unboundedness from below of the quadratic form for $m=1$ and $N$ sufficiently large cannot be considered valid. Our second main result (Theorem \\ref{ub1}) fills in this gap and we prove that for any $N\\geq 2$ the quadratic form $\\mathcal F_{\\alpha}$ is unbounded from below for $m<m^*(2)$. In analogy with the bosonic case, we expect that in such a case $H_{\\alpha}$ is not self-adjoint and all its self-adjoint extensions are unbounded from below\n\nLet us make a few remarks on the above-mentioned results. First, we emphasize that in the case $N=2$ we fully characterize stability: the model is stable for $m>m^*(2)$ and unstable $m<m^*(2)$. Whereas the latter was already found in \\cite{Sh} by means of the theory of self-adjoint extensions, the former is proved here for the first time.\n\nSecond, if $N>2$ we expect the condition $m>m^*(N)$  to be far from optimal for stability. This is due to the crucial role played by the restriction to antisymmetric wave functions (see the discussion in Section \\ref{instability: sec}) so that the system might  be stable also if our condition  is violated.  \n\nThird, the fact that the $(N\\!+\\!1)$-particle system is unstable at least when $m$ is below the \\emph{same} threshold $m^*(2)$ for the instability of the $(2\\!+\\!1)$-particle system has a rather natural interpretation: the instability of a subsystem made of two fermions plus the different particle is responsible for the instability of the whole system. \n\n\nWe want to mention that in the final stage of the preparation of this work we became aware of a recent paper  \\cite{m4} where the case of two fermions plus a different particle is studied using the theory of self-adjoint extensions. We believe that a  comparison with our methods and  results would be of great interest for the further developments of the subject.\n\nThe paper is organized as follows. In Section \\ref{main results: sec} we introduce the renormalised quadratic form $\\mathcal F_{\\alpha}$ and the STM extension $H_{\\alpha}$ and we formulate our main results. In Section \\ref{stability: sec} we give the proof  of Theorem \\ref{clbou}. In Section \\ref{instability: sec} we give the proof of Theorem \\ref{ub1}. In the Appendix we  briefly outline the formal renormalisation  procedure to derive $\\form$.\n\n\\vs\n\nFor the convenience of the reader, we collect here  some useful notation that will be used throughout the paper. We use the notation $ \\ldf(\\R^{d}) $ (resp. $ \\hf(\\R^{d}) $, $ \\hcf(\\R^{d}) $, etc.)  for the space containing totally antisymmetric functions belonging to $ L^2(\\R^{d}) $ (resp. $ H^1(\\R^{d}) $, $ H^{-1\/2}(\\R^{d}) $, etc.). We often use the short-hand notation $\\|\\cdot\\|_{L^2}$, $\\|\\cdot\\|_{\\ldf}$, etc. for the associated norms $\\|\\cdot\\|_{L^2(\\bR^d)}$, $\\|\\cdot\\|_{\\ldf(\\R^{d})}$, etc.\n\\newline\nFor a vector $\\xv \\in \\bR^3$ we set $x = |\\xv|$. Moreover we define $ \\Kv := \\lf( \\kv_2,\\ldots , \\kv_{N-1} \\ri) $ and, for $ i = 1, \\ldots, N $,\n\t\\bdm\n\t\t\\breve{\\kkv}_i := \\lf( \\kv_1, \\ldots, \\kv_{i-1}, \\kv_{i+1}, \\ldots, \\kv_N \\ri).\n\t\\edm\nFor any $f \\in L^2(\\bR^d)$ the Fourier transform is defined by $ \\hat{f}(\\kv)=(2 \\pi)^{-d\/2} \\int_{\\bR^d} \\! \\diff \\xv \\, e^{-i \\kv \\cdot \\xv} f(\\xv)\\,.$\n\\newline\nThe functions $G_{\\lambda}$,  $\\pot$, $L_{\\lambda}$, with $\\lambda >0$, and $D(\\Kv)$ are defined in \\eqref{Green function}, \\eqref{pot xi}, \\eqref{Lla} and \\eqref{Dk} respectively.\n\n\n\n\n\n\n\n\n\n\\vs\n\n\\section{Main results}\n\\label{main results: sec}\n\nIn this section we introduce the quadratic form $\\form$,  the STM extension $H_{\\alpha}$ and we formulate our main results.\n\n\n\\subsection{The quadratic form $\\form$}\n\n\n\\n\nFor a three dimensional quantum system  composed by $N$ identical fermions, with mass  one, plus a different particles, with mass $m$, with a two-body zero-range interaction    the formal many-body Hamiltonian is \n\\beq\n\t\\label{formal Hamiltonian}\n\t\\tilde{H} : = - \\frac{1}{2 m}\\Delta_{\\xv_0} - \\f{1}{2} \\sum_{i =1}^N \\Delta_{\\xv_i} + \\mu   \\sum_{i = 1}^N  \\delta(\\xv_0 - \\xvi)\\, ,\n\\eeq\nwhere $ \\xv_i \\in \\RT $, $ i = 0, \\ldots, N $, and $\\mu \\in \\bR$. Introducing the centre of mass and relative coordinates \n\\beq\n\t\\lf\\{\n \t\t\\begin{array}{ll}\n\t\t\t\\mathbf{X} : =\\displaystyle\\frac{1}{m+N} \\Big( m\\xv_0 + \\sum_{i=1}^N \\xv_i \\Big), \t&\t\\mbox{}\t\\\\\n\t\t\t\\yv_i  : = \\xv_0 - \\xv_i ,\t\t&\t\\mbox{for} \\:\\:\\: i = 1,\\ldots N\\,,\n\t\t\\eay\n\t\\ri.\n\\eeq\none obtains\n\\beq\n\t\\tilde{H} = \\hcm + \\tx\\frac{m+1}{2m}  H \n\\eeq\nhere $ \\hcm : =- [2(m+N)]^{-1} \\Delta_{\\mathbf{X}} $ and\n\\beq\n\t\\label{cm Hamiltonian}\n\tH : = - \\sum_{i=1}^N \\Delta_{\\yv_i} -  \\frac{2}{m+1} \\sum_{i < j} \\nabla_{\\yv_i}  \\cdot \\nabla_{\\yv_j} +  \\mu \\sum_{i = 1}^N \\delta(\\yvi)\\,,\n\\eeq\n$\\nabla_{\\yv_i} $ denoting the gradient with respect to $\\yv_i$.  \nWe also introduce the free Hamiltonian\n\\beq\n\t\\label{free Hamiltonian}\n\t\t\\dom(H_0) := H^2_{\\mathrm{f}}(\\bR^{3N}),\t\\hspace{1cm}\tH_0 : = - \\sum_{i=1}^N \\Delta_{\\yv_i} -  \\frac{2}{m+1} \\sum_{i < j} \\nabla_{\\yv_i}  \\cdot \\nabla_{\\yv_j}\\,,\n\\eeq\nand its restriction to functions vanishing in a neighbourhood of the hyperplanes \n$ \\{\\yv_i = 0 \\}$\n\\beq\n\t\\label{free2 Hamiltonian}\n\\dom(\\dot{H}_0)  : = \\!\\bigg\\{ \\psi \\in \\hdf(\\R^{3N}) \\: \\bigg| \\! \\int_{\\R^3} \\!\\!\\diff \\kv_i \\: \\hat\\psi(\\kv_1, \\ldots, \\kv_N) = 0 \\,, \\;\\;\\;\\;i=1,\\ldots ,N \\bigg\\},\t\\hspace{1cm}\t\\dot{H}_0 : = H_0\\big|_{\\dom(\\dot{H}_0)}\\,.\n\\eeq\n\nIn order to give a rigorous meaning to the formal expression \\eqref{cm Hamiltonian} as a self-adjoint operator in  $ \\ldf(\\R^{3N}) $, one can use Krein's theory to construct self-adjoint extensions of the operator  \\eqref{free2 Hamiltonian} (see \\cite{m3}). Here, instead, we follow a different approach and we investigate the quadratic form associated with the expectation value $ \\bra{\\psi} H \\ket{\\psi} $. However, because of the singularity of the ``potential'' $ \\delta(\\yv_i) $ the quadratic form has to be defined via a renormalisation procedure in the Fourier space. The idea  of the construction is given in the Appendix (see also  \\cite{DFT},\\cite{FT}) and here we only give   the final result.\n\n\\n\nLet us denote for any $\\lambda>0$\n\\beq\n\t\\label{Green function}\n\t\\green(\\kv_1, \\ldots, \\kv_N) : = \\bigg[ \\sum_{i=1}^N k_i^2 + \\frac{2}{m+1}\\sum_{i < j} \\kv_i \\cdot \\kv_j + \\la \\bigg]^{-1}\\,,\n\\eeq\n\\beq\\label{Lla}\nL_{\\lambda}(\\kv_1,\\ldots,\\kv_{N-1}) := 2 \\pi^2 \\bigg( \\frac{m(m+2)}{(m+1)^2} \\sum_{i = 1}^{N-1} k_i^2 + \\frac{2m}{(m+1)^2} \\sum_{i <j} \\kv_i \\cdot \\kv_j + \\la \\bigg)^{1\/2}\\,,\n\\eeq\nand, for any ``charge'' $\\xi \\in   \\hcf(\\R^{3N-3})  $,  let us  denote the ``potential'' produced by $\\xi$ in the Fourier space by\n\\beq\n\t\\label{pot xi}\n\t\\lf( \\widehat{\\pot \\xi} \\ri)(\\kv_1, \\ldots, \\kv_N) : = \\sum_{i=1}^N (-1)^{i+1} \\green(\\kv_1, \\ldots, \\kv_N) \\, \\hat\\xi( \\breve{\\kkv}_i)\\,\n\\eeq\nwhere  $\\breve{\\kkv}_i := (\\kv_1, \\ldots, \\kv_{i-1}, \\kv_{i+1}, \\ldots, \\kv_N) $.  Then the quadratic form $\\form$  in the Hilbert space $\\ldf(\\R^{3N})$ is defined as follows\n\\beq\n\t\\label{form domain}\n\t\\dom(\\form) : = \\lf\\{ \\psi \\in \\ldf(\\R^{3N}) \\: \\Big| \\: \\exists \\xi \\in \\dom(\\qform) \\: \\;\\; \\mathrm{s.t.} \\;\\; \\: \\phi^{\\la} : = \\psi - \\pot \\xi \\in \\hf(\\R^{3N}) \\ri\\},\n\\eeq\n\\beq\n\t\\label{form}\n\t\\form[\\psi] : = \\F_0[\\phi^{\\la}] + \\la \\| \\phi^{\\la} \\|_{L^2(\\R^{3N})}^2 - \\la \\lf\\| \\psi \\ri\\|_{L^2(\\R^{3N})}^2 + N \\qform\\lf[\\xi\\ri],\n\\eeq\nwhere $\\lambda >0$, $ \\F_0[\\phi] : = \\bra{\\phi} H_0 \\ket{\\phi} $ and  $\\Phi_{\\alpha}^{\\lambda}$ is the following form on the charge $ \\xi $  \n\\beq\\label{domqform}\n \\dom(\\qform) := \\hcfpiu   (\\R^{3N - 3})\\,,\n\\eeq\n\\beq\n\t\\label{qform}\n\t\\qform[\\xi] : = \\dqform[\\xi] + \\oqform[\\xi]\\,,\n\\eeq\n\\beq\t\\label{dqform}\n\t\\dqform[\\xi] : = \\int_{\\R^{3N-3}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\lf[ \\al +\t\t\n\tL_{\\lambda}(\\kv_1,\\ldots,\\kv_{N-1}) \\ri],\n\\eeq\n\\bml{\n\t\\label{oqform}\n\t\\oqform[\\xi] : = (N-1) \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kv_2 \\cdots \\diff \\kv_{N-1} \\:\t \\hat\\xi^*(\\sv, \\kv_2, \\ldots, \\kv_{N-1}) \\hat\\xi (\\tv, \\kv_2, \\ldots, \\kv_{N-1}) \\cdot \t\\\\\n\t\\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) .\n}\n\n\\vspace{0.1cm}\n\n\\n\nWe notice that for $\\xi \\in  \\dom(\\qform)$ we have $\\pot \\xi \\in  \\ldf(\\R^{3N})$ and $\\pot \\xi \\notin \\hf(\\R^{3N})$. Therefore the decomposition in \\eqref{form domain} is meaningful. Moreover, as we shall see in Section \\ref{stability: sec}, the form $\\qform$ is well-defined on $\\dom (\\qform)$.\n\nIt is worth mentioning that the fermionic constraint implies not only that the wave function is totally antisymmetric but also that the form on the charges \\eqref{qform} differs from the bosonic case by a sign in front of the off-diagonal part \\eqref{oqform}. This fact results in a weaker effective interaction among the fermions and the stability problem is qualitatively different.\n\nIn order to formulate our main results on the form $\\form$ we first  introduce our definition of stability parameter. Let us consider the following function\n\\beq\\label{Lambda}\n\\Lambda (m,N) : =   2\\pi^{-1} (N-1)    (m+1)^2 \\bigg[ \\f{1}{ \\sqrt{m(m+2)}} - \\arcsin\\bigg(\\f{1}{m+1}\\bigg) \\bigg]\\,.\n\\eeq\nIt is easy to check that, for $N$ fixed, the function $\\Lambda(m,N)$ is decreasing in $m$ and \n\\beq\n\t\\lim_{m\\rightarrow 0} \\Lambda(m,N)=\\infty,\t\\hspace{1cm}\t\\lim_{m\\rightarrow \\infty} \\Lambda(m,N)=0\\,.\n\\eeq  \nThen we have\n\n\\begin{definition}[Stability parameter $m^*(N)$]\n\t\\mbox{}\t\\\\ \nFor $N$ fixed, we define $m^*(N)$ as the unique solution to the equation \n\\beq\\label{La=1}\n\\Lambda(m,N)=1\\,.\n\\eeq\n\\end{definition}\n\\vspace{0.2cm}\n\\n\nIf we define $\\theta := \\arctan \\sqrt{1+\\f{2}{m}}$,  where $\\theta \\in (\\f{\\pi}{4}, \\f{\\pi}{2})$, a direct computation shows that equation \\eqref{La=1} can be equivalently written as\n\\beq\\label{teta}\n \\cot 2 \\theta + 2 \\theta  -\\f{\\pi}{2}\\bigg( 1 - \\f{1}{N-1} \\cos^2 2 \\theta \\bigg)=0\\,.\n\\eeq\nWe remark that   \\eqref{teta}  reduces for $N=2$ to the equation found for the critical mass in  \\cite[p. 12871, note 10]{pc}. \nWe also notice that $m^*(N)$ is positive and increasing with $N$. In particular,  the condition $\\Lambda(m,N) <1$ is equivalent to  $m>m^*(N)$. This condition is crucial to guarantee closure and boundedness from below of the form $\\form$.\n\n\\begin{theorem}[Stability for $ m > m^*(N)$]\n\\label{clbou}\n\\mbox{}\t\\\\\nLet $N\\geq 2$ and  $m>m^*(N)$.  Then  the quadratic form $\\form$ is closed and bounded from below. In particular,  it is positive for $\\alpha \\geq 0$ and \n\\beq\\label{inff}\n\\form[\\psi] \\geq -  \\f{\\alpha^2}{4 \\pi^4 \\big(1\\!-\\!\\Lambda(m,N) \\big)} \\, \\|\\psi\\|^2_{L^2},\t\\hspace{1cm} \\psi \\in \\dom(\\form)\\,,\n\\eeq\nfor $\\alpha <0$.\n\\end{theorem}\n\n\\n\nThe proof   will be given in Section \\ref{stability: sec}. \n\nConcerning the instability problem, our  result is the following.\n\n\\begin{theorem}[Instability for $ m < m^*(2) $]\n\\label{ub1}\n\\mbox{}\t\\\\\nLet $N\\geq 2$ and $m<m^*(2)$. Then the quadratic form $\\form$ is unbounded from below for any $\\alpha \\in \\bR$.\n\\end{theorem}\n\n\\n\nThis theorem generalizes the result obtained for $N=2$ in \\cite{FT} and it is proved in Section \\ref{instability: sec}. In the case $N=2$, Theorems \\ref{clbou} and \\ref{ub1} show that the system is stable for $m>m^* (2)$ and unstable for $m<m^*(2)$, therefore our analysis is complete.  When  $N>2$ the problem remains open: no rigorous result is available for $m^*(2)<m<m^*(N)$. \n\n\n\\n\n\n\\vs\n\n\\subsection{The STM extension $H_{\\alpha}$}\n\nHere we recall the standard definition of the STM extension $H_{\\alpha}$ and its connection with the quadratic form $\\form$. Moreover, as a direct consequence of Theorem \\ref{clbou},  we prove our main result on $H_{\\alpha}$. \n\nAs we mentioned in the introduction, $H_{\\alpha}$  is    a distinguished  symmetric  extension of the operator  \\eqref{free2 Hamiltonian}. For the details of the construction we refer to \\cite{m3} and here we only give  the definition\n\\bml{\n\t\\label{domain Ha 2}\n\t\\dom(H_{\\al}) : = \\bigg\\{ \\psi \\in \\ldf(\\R^{3N}) \\: \\bigg| \\: \\exists \\, \\xi \\in H^{3\/2}_{\\mathrm{f}} (\\bR^{3N-3}) \\: \\;\\mathrm{s.t.} \\;\\: \\phi_{\\la} := \\psi - \\pot \\xi \\in \\hdf(\\R^{3N}),\t\\\\\n\t\\int_{\\R^3} \\diff \\kv_i \\: \\hat\\phi_{\\la}(\\kv_1, \\ldots, \\kv_N) =  (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i), \\, i = 1, \\ldots, N \\bigg\\}\\,,\n}  \n\\beq\n\t\\label{action Ha}\n\t(H_{\\al} + \\la) \\psi = (H_0 + \\la) \\phi_{\\la}\\,,\n\\eeq\nwhere $\\lambda>0$\n and\n\\beq\t\\label{operator A}\n(\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i) \n : = (-1)^{i+1} \\bigg[ \\alpha + L_{\\lambda}(\\breve{\\kkv}_i) \\bigg] \\hat{\\xi}(\\breve{\\kkv}_i) - \\!\\!  \\sum_{j=1, j\\neq i}^N \\!\\!  (-1)^{j+1}  \\!\\!  \\int_{\\R^3} \\diff \\kv_i \\:  G_{\\lambda} (\\kv_1,\\ldots, \\kv_N) \\, \\hat{\\xi}(\\breve{\\kkv}_j) \\,.\n\\eeq\nThe last equality in \\eqref{domain Ha 2} should be understood as the boundary condition satisfied by any $\\psi \\in \\dom(H_{\\alpha})$. In fact, \nby a straightforward computation, one verifies that such equality implies  the following asymptotic condition for $R\\rightarrow \\infty$ \n\\beq\n \\int_{k_i <R} \\diff \\kv_i \\: \\hat{ \\psi}(\\kv_1,\\ldots, \\kv_N)= 4 \\pi R \\, (-1)^{i+1} \\hat{\\xi}(\\breve{\\kkv}_i) + \\alpha \\,  (-1)^{i+1} \\hat{\\xi}(\\breve{\\kkv}_i) +o(1)\n \\eeq\nand this is exactly the standard boundary condition, formulated in the Fourier space,  characterizing the STM extension (see \\cite{m3}).\n\nThe next step is to establish the connection with the form $\\form$. For any $\\psi \\in \\dom (H_{\\alpha})$ one has\n\\begin{eqnarray}\n&& \\bra{\\psi} H_{\\alpha} +\\lambda  \\ket{\\psi}  = \\bra{\\psi} H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} = \\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} + \n \\bra{\\pot \\xi } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}}\\nonumber\\\\\n &&=  \\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} +\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) \\int_{\\bR^3} \\! \\diff \\kv_i \\, \\hat{\\phi}_{\\lambda} (\\kv_1,\\ldots,\\kv_N)\\nonumber\\\\\n &&=\\bra{\\phi_{\\lambda} } H_{0} +\\lambda  \\ket{\\phi_{\\lambda}} +\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i)\\,.\n\\end{eqnarray}\n\n\n\\n\nExploiting \\eqref{operator A} and the antisymmetry of $\\hat{\\xi}$  we have\n\\bml{ \\label{afi}\n \\sum_{i=1}^N (-1)^{i+1} \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; \\hat{\\xi}^*(\\breve{\\kkv}_i) (\\mathcal A^{\\lambda}_{\\alpha,i}  \\hat\\xi ) (\\breve{\\kkv}_i) \\\\ \n = \\sum_{i=1}^N \\int_{\\bR^{3N-3}} \\!\\!\\! \\diff \\breve{\\kkv}_i \\; |\\hat{\\xi}(\\breve{\\kkv}_i)|^2 \\big[ \\alpha + L_{\\lambda}(\\breve{\\kkv}_i) \\big] \n -\\sum_{i,j=1, i\\neq j}^N \\!\\!\\!\\! (-1)^{i+j} \n \\int_{\\bR^{3N}}\\!\\! \\diff \\kv_1\\cdots \\diff \\kv_N \\,  \\hat\\xi^* (\\breve{\\kkv}_i) \\hat\\xi (\\breve{\\kkv}_j) \\, G_{\\lambda} (\\kv_1, \\ldots , \\kv_N) \\\\\n = N \\dqform [\\xi] + N(N-1) \\oqform [\\xi]\\,.\n}\nTherefore we conclude\n\\beq\\label{hfa}\n    \\bra{\\psi} H_{\\alpha} \\ket{\\psi}    =     \\form \\lf[\\psi \\ri], \t\\hspace{1cm}\t\\psi \\in \\dom(H_{\\al})\\,.\n\\eeq\n\n\\n\nFrom the above relation and Theorems \\ref{clbou}, \\ref{ub1},  we obtain the following result.\n\n\\begin{theorem}[Self-adjointness and boundedness from below]\n\\label{thhal}\n\\mbox{}\t\\\\\nOne has the  following two possible alternatives:\n\\ben[(i)]\n\t\\item If $N\\geq 2$ and $m>m^*(N)$, then $\\form$ defines a unique self-adjoint and bounded from below extension $\\hat{H}_{\\alpha}$ of the operator $H_{\\alpha}$. In particular, $\\hat{H}_{\\alpha}$ is positive for $\\alpha \\geq 0$ and \n\\beq\\label{infsp}\n\\inf \\sigma(\\hat{H}_{\\alpha}) \\geq - \\f{\\alpha^2}{4 \\pi^4 \\big(1 - \\Lambda(m,N) \\big)}, \t\\hspace{1cm}\t\\text{for} \\;\\;\\;\\alpha <0.\n\\eeq \n\t\\item  If $N \\geq 2$ and $m<m^*(2)$, then, for any $\\alpha \\in \\bR$, no self-adjoint extension $ \\hat{H}_{\\alpha} $ of $ H_{\\alpha} $ such that $ \\dom(\\hat{H}_{\\alpha}) \\subset  \\dom(\\form) $ can be both  self-adjoint and bounded from below.\n\\een\n\n\\end{theorem}\n\n\\begin{proof}\nThe assertion (i) immediately follows from Theorem \\ref{clbou} and the variational characterization of the infimum of the spectrum of a self-adjoint operator.\n\nThe converse (ii) can be obtained by exploiting the Birman-Krein-Vishik theory of positive self-adjoint extensions (see, e.g., \\cite{simon}, \\cite{P1},\\cite{P2}). We omit the details and refer to \\cite[Proposition 4.1]{FT} where an analogous results was proven.\n\\end{proof}\n\n\\vs\n\n\\n\nWe remark that, in the case $N\\geq 2$ and $m<m^*(2)$, an analogy with the bosonic case suggests that any self-adjoint extension of $H_{\\alpha}$ is unbounded from below and the Thomas effect occurs, although such a complete characterization is beyond our purposes. The most natural tool to approach the analysis of this case  is indeed the theory of self-adjoint extensions (see, e.g., \\cite{Sh},\\cite{m4}). \n\n\n\\n\nWe conclude this section with a brief comment relative to the special case $N=2$, $m>m^*(2)$. In particular we want to give the explicit characterization of our Hamiltonian $\\hat{H}_{\\alpha}$ in order to make more transparent a possible comparison of our result with the result in \\cite{m4}.\n\n\\n\nIn Section 3 we shall prove the estimate \\eqref{cC}, which in particular implies  that the form  $\\Phi^{\\lambda}_{\\alpha}$, defined on $H^{1\/2}(\\R^3)$, is closed and positive for $\\lambda$ sufficiently large. Therefore it defines a positive, self-adjoint operator $\\Gamma^{\\lambda}_{\\alpha}$, $\\dom (\\Gamma^{\\lambda}_{\\alpha})$ in $L^2(\\R^3)$ explicitly given by\n\\bdn\\label{opGa}\n\\dom(\\Gamma^{\\lambda}_{\\alpha})= \\!\\lf\\{ \\xi \\in H^{1\/2}(\\R^3)\\; \\Big| \\; \\Gamma^{\\lambda}_{\\alpha} \\hat\\xi \\in L^2(\\R^3) \\ri\\},\t\\nonumber\t\\\\\n\\lf(\\Gamma^{\\lambda}_{\\alpha} \\hat\\xi \\ri)(\\qv)\\!= [\\alpha + L_{\\lambda}(\\qv) ] \\hat\\xi(\\qv) + \\!\\!\\int_{\\R^3}\\!\\!\\! \\diff \\pv \\, G_{\\lambda}(\\pv, \\qv) \\hat\\xi(\\pv).\n\\edn\nExploiting this fact and following the same line of \\cite[Section5]{DFT}, we obtain\n\\bml{\n\t\\label{domain Ha3}\n\t\\dom(\\hat{H}_{\\al})  = \\bigg\\{ \\psi \\in \\ldf(\\R^{6}) \\: \\bigg| \\: \\exists \\, \\xi \\in \\dom(\\Gamma^{\\lambda}_{\\alpha})  \\: \\;\\mathrm{s.t.} \\;\\: \\phi_{\\la} := \\psi - \\pot \\xi \\in \\hdf(\\R^{6}),\t\\\\\n\t\\int_{\\R^3} \\diff \\pv \\: \\hat\\phi_{\\la}(\\pv, \\qv)  =  (\\Gamma^{\\lambda}_{\\alpha}  \\hat\\xi ) (\\qv) \\bigg\\}\\,,\n}  \n\\beq\n\t\\label{action Ha3}\n\t(\\hat{H}_{\\al} + \\la) \\psi = (H_0 + \\la) \\phi_{\\la}.\n\\eeq\nWe notice that the above operator differs from the operator \\eqref{domain Ha 2}, \\eqref{action Ha} only in a larger  class of admissible charges $ \\xi $, i.e., the domain $ \\dom(\\Gamma^{\\lambda}_{\\alpha}) $ strictly contains $ H^{3\/2}(\\R^3) $. We also underline that the boundary condition satisfied on the hyperplanes by an element of \\eqref{domain Ha3} is the standard STM boundary condition.\n\n\n\n\n\\section{Closure and boundedness from below of $ \\form $}\n\\label{stability: sec}\n\nThe proof of Theorem \\ref{clbou} is based on a careful estimate from below and from above of the form $\\qform$ on the charge $\\xi$.  \nIf $ N > 2 $ (with an obvious modification in the case $ N = 2 $) we rewrite \nboth the diagonal and the off-diagonal terms  of $\\Phi_{\\alpha}^{\\lambda}$, defined in  \\eqref{dqform}, \\eqref{oqform},  in a more manageable form (see \\cite{m3}), by introducing the change of coordinates\n\\bdm\n\t\\sv \\longrightarrow \\siv : = \\sv + \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i, \t\\hspace{1cm} \t\\tv \\longrightarrow \\tav : = \\tv + \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i.\t\t\n\\edm\nThen\n\\beq\n\t\\label{dqform 1}\n\t\\dqform[\\xi] = \\alpha \\, \\| \\xi\\|_{L^2(\\R^{3N-3})\n\t}^2 +\t\n\t2 \\pi^2 \\int_{\\R^{3N-3}} \\diff \\sigma \\diff \\Kv \\:\n\t\\big| \\xit(\\siv, \\Kv\n\t) \\big|^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} \\sigma^2 + \\dk+ \\la}\\,,\n\\eeq\n\\beq\n\t\\label{oqform 1}\n\t\\oqform[\\xi] = (N-1) \\int_{\\R^{3N}} \\diff \\siv \\diff \\tav \\diff \\Kv \\:\n\t\\xit^*(\\siv, \\Kv\n\t) \\: \\xit (\\tav, \\Kv\n\t)\n\t\\lf( \\sigma^2 + \\tau^2 + \\tx\\frac{2}{m+1} \\siv \\cdot \\tav + \\dk + \\la \\ri)^{-1}\\,,\n\\eeq\nwhere\n\\beq\n\\Kv := \\kv_2, \\ldots , \\kv_{N-1}\\,,\n\\eeq\n\\beq\n\t\\label{xit}\n\t\\xit \\lf(\\siv, \\Kv\n\t\\ri) := \\hat\\xi\\bigg(\\siv - \\frac{1}{m+2} \\sum_{i=2}^{N-1} \\kv_i,  \\kv_2, \\ldots, \\kv_{N-1}\n\t\\bigg)\\,,\n\\eeq\n\\beq\n\t\\label{Dk}\n\t\\dk : =\n\t\\frac{m}{(m+1)(m+2)} \\bigg( (m+3) \\sum_{i=2}^{N-1} k_i^2 + 2 \\sum_{i < j} \\kv_i \\cdot \\kv_j \\bigg)\\,.\n\\eeq\nNotice that $\\dk$ satisfies the bound (see \\eqref{el12} in the following)\n\\beq\n\t\\label{estimate Dk}\n\t\\frac{m}{m+1} \\sum_{i=2}^{N-1} k_i^2 \\leq \\dk \\leq \\frac{m(m+N+1)}{(m+1)(m+2)} \\sum_{i=2}^{N-1} k_i^2\\,.\n\\eeq\nLast, setting \n\\beq\n\t\\siv := \\sqrt{\\dk + \\la} \\: \\pv,\t\\hspace{1cm}\t\\tav : = \\sqrt{\\dk + \\la} \\: \\qv\\,,\n\\eeq\nand\n\\beq\n\t\\label{charge Qk}\n\tQ_{\\kkv}(\\pv) : = (\\dk + \\la)^{3\/4} \\: \\xit \\lf(\\sqrt{\\dk + \\la} \\: \\pv, \\Kv\n\t\\ri)\\,,\n\\eeq\nwe obtain\n\\beq\n\t\\label{qform 1}\n\t\\Phi_0^{\\lambda}[\\xi]  = \\qform[\\xi] -  \\alpha \\| \\xi \\|_{L^2(\\R^{3N-3})}^2 =  \\int_{\\R^{3N-6}} \\diff \\K\n\t\\sqrt{\\dk \\!+\\! \\la} \\;\\,  F_1 \\! \\lf[ Q_{\\kkv} \\ri]\\,,\n\\eeq\nwhere for any $\\zeta \\geq 0$ we introduced the quadratic form in $L^2(\\R^3)$\n\\beq\n\t\\label{fform}\n\t\\dom(F_{\\zeta})\\!:=\\!\\dom(F_1)\\!= \\! \\bigg\\{ \\! f \\! \\in \\! L^2(\\R^3)\\,\\bigg|\\! \\int_{\\R^3} \\!\\! \\diff \\pv \\sqrt{p^2 \\!+\\!1} \\, |f(\\pv)|^2  <\\infty \\bigg\\},\t\\quad\tF_{\\zeta} \\lf[f\\ri\n\t: = \\dform_{\\zeta} \\lf[f\\ri]\n\t+ \\oform_{\\zeta}\\lf[f\\ri]\n\\eeq\nand \n\\beq\n\t\\label{dform}\n\t\\dform_{\\zeta} \\lf[f \\ri]\n\t= 2 \\pi^2 \\int_{\\R^{3}} \\diff \\pv  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} p^2 + \\zeta} \\: \\lf| f\n\t(\\pv) \\ri|^2 \\,,\n\\eeq\n\\beq\n\t\\label{oform}\n\t\\oform_{\\zeta} \\lf[f \\ri]\n\t=  (N-1) \\int_{\\R^{6}} \\diff \\pv \\diff \\qv \\,\t\\frac{f\n\t^*(\\pv) f\n\t(\\qv)}{  p^2 + q^2 + \\tx\\frac{2}{m+1} \\pv \\cdot \\qv + \\zeta}\\,   .\n\\eeq\n\n\\vspace{0.3cm}\n\\n\nUsing the representation \\eqref{qform 1}, \\eqref{fform}, \\eqref{dform}, \\eqref{oform} we obtain the following estimate for $\\Phi_0^{\\lambda}$, which is the crucial ingredient for the proof of Theorem \\ref{clbou}.   \n\n\n\\begin{proposition}[Upper and lower bounds for $ \\Phi_0^{\\lambda} $]\n\\label{stiqform}\n\\mbox{}\t\\\\\nFor any $\\xi \\in \\dom(\\qform)$ we have\n\\beq\\label{abqform}\n\\Big(1-\\Lambda(m,N) \\Big)\\; \\Phi_{0,\\lambda}^{\\mathrm{diag}}  [\\xi] \\;  \\leq \\;  \\Phi_0^{\\lambda} [\\xi]  \\; \\leq \\; \\Big( 1 + \\Gamma(m,N)\\Big) \\;\\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\\,, \n\\eeq\nwhere\n\\beq\\label{Gamma}\n\\Gamma(m,N) : = \\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri)\\,.\n\\eeq\n\\end{proposition}\n\n\n\\vspace{0.3cm}\n\nThe proof  is based on a careful analysis of the form $F_{\\zeta}$, $\\zeta \\in [0,1]$,  reduced to   each subspace with fixed  angular momentum $l$ and it is postponed to the end of this section. First we introduce some useful  notation and we prove some preliminary lemmas.\n\nFor any $f\\in L^2(\\R^3)$ we consider the  expansion\n\\beq\\label{exsh}\nf(\\pv) = \\sum_{l=0}^{\\infty}\\sum_{m=-l}^l f_{lm}(p) Y_{l}^m(\\theta_p, \\phi_p)\\,,\n\\eeq\nwhere $\\pv=(p, \\theta_p,\\phi_p)$ in spherical coordinates and $Y_{l}^m$ denotes the spherical harmonics of order $l,m$. We notice that $f\\in \\dom(F_1)$ is equivalent to\n\\beq\n\\sum_{l=0}^{\\infty}\\sum_{m=-l}^l \\int_0^{\\infty}\\!\\!dp\\, p^2 \\sqrt{p^2 +1}\\,  \\lf|f_{lm}(p)\\ri|^2 <\\infty\\,.\n\\eeq\nMoreover, we denote by $P_l$ the Legendre polynomial of order $l=0,1,\\ldots$ explicitly given by\n\\beq\\label{leg}\nP_l(y) = \\f{1}{2^l l!} \\f{d^l}{dy^l} (y^2 -1)^l \\,,\t\\hspace{1cm}\t y \\in [-1,1]\\,.\n\\eeq\n In the first lemma we decompose $F_{\\zeta}$ in each subspace of fixed angular momentum $l$.\n\n\\begin{lemma}[Decomposition of $ F_{\\zeta} $]\n\\label{lemma0}\n\\mbox{}\t\\\\\nFor $f\\in \\dom(F_{1})$ we have\n\\beq\t\\label{Fzl}\nF_{\\zeta}\\lf[f \\ri] \\; =\\;  \t\\sum_{l=0}^{\\infty}\\sum_{m=-l}^l  G_{\\zeta,l} \\lf[ f_{lm} \\ri] \\;  =:  \\; \\sum_{l=0}^{\\infty}\\sum_{m=-l}^l  \\lf( G^{\\mathrm{diag}}_{\\zeta} \\lf[f_{lm} \\ri] + G^{\\mathrm{off}}_{\\zeta,l} \\lf[f_{lm} \\ri]  \\ri)\\,,\n\\eeq  \nwhere for $g \\in L^2((0,\\infty), \\, p^2 \\sqrt{p^2+1}\\,dp)$\n\\begin{eqnarray}\n\t\\label{G diag off}\n&&G^{\\mathrm{diag}}_{\\zeta} \\lf[ g \\ri] : = 2 \\pi^2 \\!\\! \\int_0^{\\infty} \\!\\!\\! dp \\, p^2 \\sqrt{\\frac{m(m+2)}{(m+1)^2} p^2 +\\zeta}\\, \\, |g(p)|^2\\,, \\\\\n&& G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] : = 2\\pi (N-1) \\!\\int_0^{\\infty} \\!\\!\\!\\! \\!dp\\!\\!\\int_0^{\\infty}\\!\\!\\!\\!\\! dq\\, p^2 g^*(p) \\, q^2 g(q) \\!\\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}\\,.\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nFor a given $f\\in \\dom(F_{1})$ we consider the expansion \\eqref{exsh}. From \\eqref{dform} we see that  $\\dform_{\\zeta}[f]= \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l G^{\\text{diag}}_{\\zeta}[f_{lm}]$.  Concerning the off-diagonal term \\eqref{oform}, \nwe denote by $\\theta_{pq}$ the angle between the vectors $\\pv$ and $\\qv$ and we consider the following expansion in Legendre polynomials:\n\\begin{eqnarray}\\label{exle}\n&&\\f{1}{p^2 +q^2 + \\f{2}{m+1} pq \\cos \\theta_{pq} +\\zeta} = \\sum_{l=0}^{\\infty}\\f{2l+1}{2} \\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}  \\,\\, P_l(\\cos \\theta_{pq})\\nonumber\\\\\n&&= \\sum_{l=0}^{\\infty} 2\\pi  \\int_{-1}^1 \\!\\!\\!\\! dy\\, \\frac{P_l(y)}{p^2 + q^2 +\\frac{2}{m+1} p q y +\\zeta}  \\sum_{m=-l}^{l} Y_{l}^{m *} (\\theta_p,\\phi_p) Y_{l}^m(\\theta_q,\\phi_q)\\,.\n\\end{eqnarray}\nIn the last line we used the addition formula for spherical harmonics (see, e.g., \\cite[Eq. (8.814)]{GR}). From (\\ref{exle}) we obtain $\\oform_{\\zeta}[f]=   \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l G^{\\text{off}}_{\\zeta,l}[f_{lm}]$.\n\\end{proof}\n\n\nIn the next lemma we give a new representation of $G^{\\text{off}}_{\\zeta,l}$ which is particularly useful to  control $G^{\\text{off}}_{\\zeta,l}$ in terms of $G^{\\text{off}}_{0,l}$ for any $\\zeta>0$. \n\n\\begin{lemma}[Estimates for $ G^{\\mathrm{off}}_{\\zeta,l}$]\n\\label{lemma01}\n\\mbox{}\t\\\\\nThe form $G^{\\mathrm{off}}_{\\zeta,l}$ can be written as\n\\begin{eqnarray}\\label{rapof}\n&&G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] = \\sum_{k=0}^{\\infty} B_{l,k} \\int_0^{\\infty}\\!\\!\\! d\\nu \\, \\nu^k e^{-\\zeta \\nu} \\lf| \\int_0^{\\infty}\\!\\!\\! dp \\, g(p) \\,p^{2+k} e^{-\\nu p^2} \\ri|^{2}\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{Blk}\n&&B_{l,k}=\n\t\t\\begin{cases}\n\t\t\t \\f{2\\pi (N-1)}{2^l l! \\,k!}  \\lf(\\! \\frac{-2}{m\\!+ \\!1} \\!\\ri)^{\\!k}  \\disp\\int_{-1}^1\\!\\!\\! dy \\, (1-y^2)^l \\frac{d^l}{dy^l} y^k \t& \\mbox{if} \\:\\:\\:\\: l \\leq k\\,,\t\\\\\n\t\t\t0\t&\t\\mbox{otherwise}\\,.\n\t\t\\end{cases}\n\\end{eqnarray}\n\n\\n \nMoreover for any $\\zeta > 0$ we have\n\\begin{eqnarray}\\label{stle}\n&&0\\leq  G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] \\leq G^{\\mathrm{off}}_{0,l} \\lf[ g\\ri] \\hspace{1cm} \\text{for $l$ even}\\,,\\\\\n&&\\nonumber\\\\\n&&G^{\\mathrm{off}}_{0,l} \\lf[ g\\ri] \\leq G^{\\mathrm{off}}_{\\zeta,l} \\lf[ g\\ri] \\leq 0 \\hspace{1cm} \\text{for $l$ odd}\\,.\\label{stlo}\n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof}\nUsing the expansion\n\\beq\n\\f{1}{p^2+q^2+ \\f{2}{m+1} pqy + \\zeta}=\n \\f{1}{p^2+q^2+\\zeta}  \n\\sum_{k=0}^{\\infty} \\lf(\\!  \\f{-2}{m+\\!1}\\, \\f{pqy}{p^2+q^2 +\\zeta} \\!\\ri)^{\\!\\!k}\n\\eeq\nand formula (\\ref{leg}), we obtain\n\\beq\\label{exse1}\nG^{\\mathrm{off}}_{\\zeta,l}\\lf[ g \\ri]= \\f{2\\pi (N-1)}{ 2^l l! } \\!\\sum_{k=0}^{\\infty}  \\lf(\\! \\f{-2}{m+1} \\!\\ri)^{\\!\\!k} \\!\\! \\int_0^{\\infty}\\!\\!\\! \\!dp \\!\\int_0^{\\infty}\\!\\!\\!\\! dq\\, \\f{p^{2+k} g^*(p) q^{2+k} g(q)  }{(p^2+q^2+\\zeta)^{k+1}}\\!\n\\int_{-1}^1\\!\\!\\!\\!dy\\, y^k \\f{d^l}{dy^l} (y^2-1)^l \\,.\n\\eeq\nIntegrating  by parts $l$ times we find\n\\begin{eqnarray}\\label{exse2}\n&&G^{\\mathrm{off}}_{\\zeta,l}\\lf[ g \\ri]=\\sum_{k=0}^{\\infty} B_{l,k} \\, k! \\!\\int_0^{\\infty}\\!\\!\\! \\!dp \\!\\int_0^{\\infty}\\!\\!\\!\\! dq\\, \\f{p^{2+k} g^*(p) q^{2+k} g(q)  }{(p^2+q^2+\\zeta)^{k+1}}\\,.\n\\end{eqnarray}\nFinally we use the identity\n\\beq\\label{iden}\n\\f{k!}{(p^2+q^2+\\zeta)^{k+1}} = \\int_0^{\\infty}\\!\\!\\!\\! d\\nu \\, \\nu^{k} \\,e^{-(p^2+q^2+\\zeta)\\nu}\n\\eeq\nin (\\ref{exse2}) and we obtain (\\ref{rapof}). Let us fix $l$ even. Then the integral in (\\ref{Blk}) is different from zero only if $k$ is even and this implies that $G^{\\mathrm{off}}_{\\zeta,l}$ is positive and the estimate (\\ref{stle}) holds. Analogously, when $l$ is  odd  the integral in (\\ref{Blk}) is different from zero only if $k$ is odd and therefore $G^{\\mathrm{off}}_{\\zeta,l}$ is negative and (\\ref{stlo}) holds.\n\\end{proof}\n\n\n\n\n\n\n\n\\noindent\nNow we study the form $G_{0,l} = G^{\\text{diag}}_{0} + G^{\\text{off}}_{0,l}$ and we show that it can be diagonalized for each $l$. \n\\begin{lemma}[Diagonalization of $G_{0,l}$]\n\\label{lemma1}\n\t\\mbox{}\t\\\\\n\t For any $g \\in L^2((0,\\infty), p^2\\sqrt{p^2+1}\\, dp)$ we have\n\\begin{eqnarray} \\label{formula0}\n&&G^{\\mathrm{diag}}_0 [g] =   2 \\pi^2 \\f{\\sqrt{m(m+2)}}{m+1}  \\int_{\\R} \\diff k \\,  \\big|  g^{\\sharp} (k) \\big|^2\\,, \\\\\n&&G^{\\mathrm{off}}_{0,l}[g] = \\int_{\\R} \\diff k \\, S_l (k) \\big|  g^{\\sharp} (k) \\big|^2\\,,\n\\label{formula1} \n\\end{eqnarray}\nwhere\n\\beq \\label{formula2}\n g^{\\sharp} (k) : = \\frac{1}{\\sqrt{2\\pi}} \\int_{\\R} \\diff x \\, e^{-ikx}\\, e^{2x} \\, g(e^x)\n\\eeq\nand \n\\beq \\label{formula3}\nS_l (k) = 2\\pi^2 (N-1) \\int_{-1}^1 \\diff y \\, P_l (y) \n\\frac{\\sinh \\lf( k \\arccos \\frac{y}{m+1} \\ri) }{ \\sin  \\lf(  \\arccos \\frac{y}{m+1} \\ri) \\, \\sinh(\\pi k)}\\,.\n\\eeq\n\\end{lemma}\n\\begin{proof}\nThe proof of (\\ref{formula0}) is straightforward. To prove (\\ref{formula1}) we first introduce the new integration variables\n $p=e^{x_1}$ and $q=e^{x_2}$, so that  the form reads\n\\begin{equation}\\label{for2}\n\\begin{split}\nG^{\\mathrm{off}}_{0,l} \\lf[g \\ri]\\;&=\\;2 \\pi(N-1) \n\\int_{\\R}  dx_1 dx_2\\,\\,e^{3 x_1}{g}^* (e^{x_1})e^{3 x_2} g(e^{x_2})\n\\int_{-1}^1 dy  \\f{  P_l(y)   }{ e^{2 x_1} + e^{2x_2} +\\f{2y}{m+1} e^{x_1 +x_2}  } \\\\\n& = \\; \\pi(N-1) \n\\int_{\\R}  dx_1 dx_2\\,\\,e^{2 x_1}{g}^* (e^{x_1})e^{2 x_2} g(e^{x_2})\n\\int_{-1}^1 dy  \\f{  P_l(y)   }{ \\cosh (x_1-x_2) +\\f{y}{m+1}  }\\,.\n\\end{split}\n\\end{equation}\nThe kernel in (\\ref{for2}) is a convolution kernel and therefore it can be diagonalized by means of the Fourier transform. Using the explicit Fourier transform of the kernel (see, e.g., \\cite{erdely}) we finally arrive at  \\eqref{formula1}.\n\\end{proof}\n\n\nOwing to the previous lemma, the problem of finding  bounds for  the form $G^{\\text{off}}_{0,l} $ is reduced to finding \nbounds for the function $S_l(k)$.\nTaking into account the identity $\\arccos z = \\f{\\pi}{2} - \\arcsin z$ and the parity of $P_l$,  we represent $S_l(k)$ as\n\\begin{equation}\\label{reps}\nS_l(k) =\n\\begin{cases}\n   \\displaystyle  - \\pi^2 (N-1) \n\\int_{-1}^1 dy \\, P_l(y) \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)\\sinh \\lf(\\f{\\pi}{2} k\\ri) \\, } & \\text{for} \\;  l \\text{ odd}, \\\\\n  \\displaystyle  \\pi^2(N-1) \n\\int_{-1}^1 dy  \\, P_l(y) \\f{\\cosh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)\\cosh \\lf(\\f{\\pi}{2} k\\ri) \\, } & \\text{for} \\;  l \\text{ even}\\,.\n\\end{cases}\n\\end{equation}\nIt is evident from this representation that $S_l (k)$ is for any $l \\geq 0 $ an even $C^{\\infty}$-function of $k$ with $\\lim_{k \\rightarrow \\infty} S_l (k)=0$.   \nBefore discussing upper and lower bound of $S_l (k) $, we shall prove the following elementary lemma.\n\\begin{lemma}[Taylor expansions of $ \\wt S^{\\mathrm{o}}_k  $ and $ \\wt S^{\\mathrm{e}}_k $]\n\\label{forever}\n\t\\mbox{}\t\\\\\nFor any fixed $k\\geq 0$ the following functions \n\\beq\\label{s0y}\n\\wt S^{\\mathrm{o}}_k (y) = \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) \\, } \\,,\t\\hspace{1cm}\t \\wt S^{\\mathrm{e}}_k (y) = \\f{\\cosh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) \\, }\n\\eeq\nhave a Taylor expansion in the variable $y \\in [-1,1]$ with positive coefficients.\n\\end{lemma}\n\\begin{proof}\nLet $\\cP$ be the set of functions whose Taylor expansion has positive coefficients.\nFirst note that $\\arcsin y$, $\\sinh y$ and $\\cosh y$ belong to $\\cP$. \nThe derivative is a linear automorphism of $\\cP$ and therefore \n\\[\n\\frac{d}{dy} \\arcsin y = \\frac{1}{\\sqrt{1-y^2} } = \\frac{1}{\\cos \\arcsin y} \\in \\cP\\,.\n\\]\nMoreover, $\\cP$ is invariant under dilations, multiplications and compositions of functions in $\\cP$.\nThus, $\\wt S^{\\mathrm{o}}_k$ and  $\\wt S^{\\mathrm{e}}_k $ belong to $ \\cP$.\n\\end{proof}\n\n\\n\nIn the next lemma we compute  lower and upper bounds for $S_l (k) $.\n\\begin{lemma}[Bounds for $S_l (k) $]\n \\label{lem4}\n\t\\mbox{}\t\\\\\nFor  any $k \\in \\R$ we have\n\\begin{eqnarray}\\label{boe}\n0  \\; \\leq  \\;           S_l(k)  \\;  \\leq     \\;   2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf(\\f{1}{m\\!+\\!1}\\ri) &\t\\hspace{0,7cm}\t  & \\text{for $l$ even}\t\\\\\n\\label{boo}\n-4\\pi (N\\!-\\!1) (m\\!+\\!1) \\bigg[ 1- \\sqrt{m(m\\!+\\!2)} \\, \\arcsin  \\lf(\\f{1}{m\\!+\\!1}\\ri) \\bigg]\n\\; \\leq S_l(k)  \\;  \\leq 0 &\t\\hspace{0,7cm} & \\text{for $l$ odd}\\,.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nLet us prove \\eqref{boo} first. The upper bound follows from \\eqref{stlo} and \\eqref{formula1}. \nAs for the lower bound,  by means of \\eqref{s0y}  we write\n\\beq \\label{tonno}\nS_l (k) = - \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    }\\int_{-1}^{+1} \\diff y \\, P_l (y) \\, \\wt S^{\\mathrm{o}}_k (y) \\,.\n\\eeq\nThe first step is to  prove that $S_l (k)$ is an increasing function of $l$ for any  fixed $k$. \nFrom \\eqref{tonno},  using \\eqref{leg} and  integrating by parts, we obtain\n\\begin{align} \\label{tonno2}\nS_{l+2}(k) &= - \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\frac{d^{l+2}}{dy^{l+2} }(y^2-1)^{l+2}  \\wt S^{\\mathrm{o}}_k (y) \\no \\\\\n& = \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\frac{d^{2}}{dy^{2} }(y^2-1)^{l+2} \\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }(y) \\no \\\\\n& = \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l+2} (l+2)!}\n\\int_{-1}^{+1} \\diff y \\, \\lf[(l+2)(l+1)(y^2-1)^{l}4y^2 + 2(l+2)(y^2-1)^{l+1}    \\ri] \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y) \\no \\\\\n&=  \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\frac{1}{2^{l} l!}   \n\\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l} \n\\frac{d^{l} \\wt S_k}{dy^{l} }  (y) + \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\f{1\\!+2(l\\!+\\!1)}{2^{l\\!+\\!1} (l\\!+\\!1)!} \\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l+1} \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y)\n\\no\\\\\n&= S_l(k)  + \\frac{\\pi^2 (N\\!-\\!1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    } \\f{1\\!+2(l\\!+\\!1)}{2^{l\\!+\\!1} (l\\!+\\!1)!} \\int_{-1}^{+1} \\!\\!\\!\\diff y \\, (y^2-1)^{l+1} \n\\frac{d^{l} \\wt S^{\\mathrm{o}}_k}{dy^{l} }  (y) \\,. \n\\end{align}\nBy Lemma \\ref{forever}, and taking into account that $l+1$ is even, we deduce that the last integral in \\eqref{tonno2} is positive and therefore  we conclude  $S_{l+2} (k) \\geq S_l(k)$. \nThis means that it is sufficient to find the minimum of $S_1 (k)$.\nFrom  \\eqref{reps} we have \n\\beq \\label{shotgun}\n S_1 (k) = \n-2\\pi^2 (N-1) \n\\int_{0}^1 \\diff y \\, \\f{y}{\\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri)}   \\f{\\sinh \\lf( k \\arcsin  \\f{ y }{m+1}\\ri) }{\\sinh \\lf(\\f{\\pi}{2} k\\ri) \\, }\\,.\n\\eeq\nWe know that $S_1(0)<0$ and $\\lim_{k \\rightarrow \\infty} S_1(k)=0$. Moreover,  the derivative of $S_1(k)$ does not vanish for $k>0$, which follows from the fact that the derivative of the function\n\\[\n\\f{\\sinh ak}{\\sinh bk}\\,,\t\\hspace{1cm} 0<a<b,\n\\]\ndoes not vanish for $k>0$. Therefore, $S_1(k)$ is monotone increasing when $k>0$ and attains    its minimum at $k=0$. Thus,\n\\begin{equation}\\label{S10}\n\\begin{split}\nS_l (k) \\;\\geq\\; S_1 (k) \\;\\geq\\; S_1 (0)\\;&=\\;-4\\pi (N-1) \\int_{0}^1 \\diff y \\, y\n\\f{ \\arcsin  \\f{ y }{m+1} }{ \\cos \\lf(  \\arcsin  \\f{ y }{m+1}\\ri) } \\\\\n&=\\;-4\\pi (N-1) (m+1)^2 \\int_0^{\\arcsin \\f{ 1 }{m+1} } \\diff z \\, z \\sin z  \\\\\n&=\\;-4\\pi (N-1) (m+1) \\bigg[1- \\sqrt{m(m+2)} \\, \\arcsin \\lf(\\f{1}{m+1}\\ri) \\bigg]\n\\end{split}\n\\end{equation}\nand \\eqref{boo} is proved. The proof of \\eqref{boe}  is completely analogous. In this case the lower bound follows from \\eqref{stle} and \\eqref{formula1}.  Using the representation\n\\beq\nS_l (k) =  \\frac{\\pi^2 (N-1)}{ \\sinh \\lf(\\f{\\pi}{2} k\\ri)    }\\int_{-1}^{1} \\diff y \\, P_l (y) \\, \\wt S^e_k (y)\n\\eeq\nobtained from \\eqref{s0y}, one sees that $S_{l+2}(k) \\leqslant S_l(k)$ and therefore  it is enough to consider $S_0(k)$. Since\n\\beq\nS_0(0)= \\pi^2 (N\\!-\\!1) \\!\\int_{-1}^{1}\\!\\!\\! \\diff y \\f{1}{\\cos \\left(\\! \\arcsin \\f{y}{m+1} \\! \\right)} = 2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) >0\\,,\n\\eeq\n$\\lim_{k \\rightarrow \\infty} S_0(k)=0$, and the derivative of $S_0(k)$ does not vanish for $k>0$, we deduce that $S_0(0)$ is the maximum of $S_0(k)$.\n\\end{proof}\n\n\nUsing the results of the previous lemmas we can finally \nprove Proposition \\ref{stiqform}. \n \n\\begin{proof}[Proof of Proposition \\ref{stiqform}]  We prove first the estimate from below in \\eqref{abqform}. From Lemmas \\ref{lemma0},  \\ref{lemma01}, \\ref{lemma1}, \\ref{lem4} we have\n\\begin{equation}\\label{sotto}\n\\begin{split}\n\\oform_1[f]\\;&=\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ even}}}\n G^{\\mathrm{off}}_{1,l} \\lf[f_{lm} \\ri]   + \\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}}\n G^{\\mathrm{off}}_{1,l} \\lf[f_{lm} \\ri] \\;\\geq\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}}\n G^{\\mathrm{off}}_{0,l} \\lf[f_{lm} \\ri] \\;=\\;\\sum_{\\substack{l,m \\\\ l\\textrm{ odd}}} \\int_{\\R} dk\\, S_l(k) \\, \\big| f^{\\sharp}_{lm} (k) \\big|^2 \\\\\n&\\geq\\; -4\\pi (N\\!-\\!1) (m\\!+\\!1)  \\bigg[ 1 - \\sqrt{m(m+2)} \\, \\arcsin \\lf( \\f{1}{m+1} \\ri) \\bigg]\n  \\;  \\sum_{l,m} \\int_{\\R} dk\\,  \\big| f^{\\sharp}_{lm} (k) \\big|^2 \\\\\n&=\\;- \\Lambda(m,\\!N) \\, \\f{2 \\pi^2 \\sqrt{m(m+2)}}{m+1}  \\sum_{l,m} \\int_{\\R} dk\\,  \\big| f^{\\sharp}_{lm} (k) \\big|^2\n\\end{split}\n\\end{equation}\nwhere $\\Lambda(m,\\!N)$ is defined in \\eqref{Lambda}. Then, by \\eqref{formula0},\n\\beq\\label{sotto1}\n\\oform_1[f] \\geq -\\Lambda(m,\\!N) \\! \\sum_{l,m} G^{\\mathrm{diag}}_{0} [f_{lm}]  \\geq -\\Lambda(m,\\!N) \\! \\sum_{l,m} G^{\\mathrm{diag}}_{1} [f_{lm} ] =  -\\Lambda(m,\\!N) \\, \\dform_1[f]\\,.\n\\eeq\nFrom the representation \\eqref{qform 1} and from \\eqref{sotto1} we have\n\\beq\n\\Phi_0^{\\lambda} [\\xi] \\geq \\big(1-\\Lambda(m,N)\\big) \\int_{\\bR^{3N-6}} \\!\\!\\! \\diff \\Kv \\, \\sqrt{D(\\Kv) +\\lambda} \\, F_1^{\\mathrm{diag}} [Q_{\\Kv}] = \\big(1-\\Lambda(m,N)\\big)\\, \\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\n\\eeq\nand the lower bound is proved. An analogous proof yields the upper bound in \\eqref{abqform}. We have\n\\begin{equation}\\label{sopra}\n\\begin{split}\n\\oform_1[f]\\;&\\leq\\;2 \\pi^2 (N\\!-\\!1)(m\\!+\\!1) \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} \\int\\!\\! dk\\,  | f^{\\sharp}_{lm} (k)|^2 \\\\\n&=\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf(\\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} G^{\\mathrm{diag}}_{0} [f_{lm}] \\\\\n&\\leq\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\sum_{l,m} G^{\\mathrm{diag}}_{1} [f_{lm}] \\\\\n&=\\;\\f{(N\\!-\\!1) (m\\!+\\!1)^2}{\\sqrt{m(m\\!+\\!2)}} \\arcsin \\lf( \\f{1}{m\\!+\\!1} \\ri) \\dform_1[f]\n\\end{split}\n\\end{equation}\nwhich, together with \\eqref{qform 1}, yields the upper bound for $\\Phi_0^{\\lambda}$. \n\\end{proof}\n\n\nLet us briefly comment on the result of Proposition \\ref{stiqform}. By means of the elementary estimate\n\\beq\\label{el12}\n-\\f{1}{2} \\sum_{i=1}^{N-1} k_i^2 \\leq \\sum_{i<j} \\kv_i \\cdot \\kv_j \\leq \\f{N-2}{2} \\sum_{i=1}^{N-1} k_i^2\\,,\n\\eeq\nwe find\n\\bml{\\label{fiup}\n\t\\qform [\\xi] = \\alpha \\|\\xi\\|^2_{L^2} +\\Phi_0^{\\lambda} [\\xi]  \\leq \\alpha \\|\\xi\\|_{L^2} + \\Big( 1 + \\Gamma(m,N)\\Big) \\;\\Phi_{0,\\lambda}^{\\mathrm{diag}} [\\xi]\t\\\\\n\t\\leq \\int_{\\bR^{3N-3}} \\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\! \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\! \\bigg[ \\al \\!+\\! 2 \\pi^2  \\Big(\\! 1\\! +\\! \\Gamma(m,N)\\Big)  \\bigg( \\frac{m(m\\!+\\!N)}{(m+1)^2} \\sum_{i = 1}^{N-1} k_i^2 + \\la \\bigg)^{\\!\\!1\/2}  \\bigg]\n}\nand\n\\beq\\label{fido}\n \\qform [\\xi] \\geq \\int_{\\bR^{3N-3}} \\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\! \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\! \\bigg[ \\al + 2 \\pi^2  \\Big(\\! 1 - \\Lambda(m,N)\\Big)  \\bigg( \\frac{m}{m\\!+\\!1} \\sum_{i = 1}^{N-1} k_i^2 + \\la \\bigg)^{\\!\\!1\/2}  \\bigg]\\,.\n\\eeq \nFrom \\eqref{fiup}, \\eqref{fido}, choosing $\\lambda$ sufficiently large if $\\alpha <0$, we conclude   \n\\beq\\label{cC}\nc\\, \\Big( 1- \\Lambda(m,N) \\Big)  \\|\\xi \\|_{\\hcfpiu} \\leq \\; \\qform[\\xi] \\; \\leq C \\;  \\|\\xi\\|_{\\hcfpiu}\\,,\n\\eeq\nwhere $c, C$ are two positive constants. Estimate  \\eqref{cC}  implies that  $\\qform[\\xi]$ is finite  for any  $\\xi \\in \\dom(\\qform)$ and therefore our definition of the quadratic forms \\eqref{form} and \\eqref{qform} is well-posed. \nOn the other hand, if in addition we assume $\\Lambda(m,N) <1$, from  \\eqref{cC}  we conclude that $\\qform$ defines a norm equivalent to the $H^{1\/2}$-norm. This is the crucial ingredient for the proof of Theorem \\ref{clbou}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{clbou}]  From \\eqref{form} and \\eqref{fido} we have\n\\bml{\n\\form [\\psi] \\geq -\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}} + N \\qform [\\xi] \\geq -\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}}\t\\\\\n + N \\!\\! \\int_{\\bR^{3N-3}} \\!\\!\\!\\! \\! \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\big| \\hat\\xi(\\kv_1, \\ldots, \\kv_{N-1}) \\big|^2 \\bigg[ \\al + 2 \\pi^2  \\Big(\\! 1 - \\Lambda(m,N) \\!\\Big)  \\bigg(\\! \\frac{m}{m\\!+\\!1} \\sum_{i = 1}^{N-1}\\! k_i^2 + \\la \\! \\bigg)^{\\!\\!1\/2}  \\bigg] \n}\nfor any $\\psi \\in \\dom(\\form)$ and $\\lambda >0$. Therefore, if $\\alpha \\geq 0$ the form $\\form$ is positive and if $\\alpha<0$ the lower bound \\eqref{inff} holds. Let us now prove that $\\form$ is closed. We choose \n\\beq\n\\lambda >\n\t\\begin{cases} \n\t\t0\t& \\text{if} \\;\\; \\alpha \\geq 0, \t\\\\\n\t\t\\f{\\alpha^2}{4 \\pi^4 \\big(1-\\Lambda(m,N)\\big) }\t&\t\\text{if} \\;\\; \\alpha <0\\,,\n\t\\end{cases}\n\\eeq\nand consider the form $\\form^{\\lambda}[\\psi] :=\\form [\\psi] +\\lambda \\|\\psi\\|^2_{L^2_{\\mathrm{f}}}$ defined on $\\dom(\\form)$. Let $\\{\\psi_n \\}$ be a sequence in $\\dom(\\form)$ such that\n\\beq\n\\lim_{n \\rightarrow \\infty} \\|\\psi_n - \\psi \\|_{L^2_{\\mathrm{f}}} =0\\,, \\hspace{1cm}\t\\lim_{n,m \\rightarrow \\infty} \\form^{\\lambda}[\\psi_n - \\psi_m] =0\\,,\n\\eeq\nwhere $\\psi \\in L^2_{\\mathrm{f}}(\\bR^{3N})$. From the definition of $\\dom(\\form)$ and $\\form^{\\lambda}$  (see \\eqref{form}, \\eqref{form domain}) we have $\\psi_n = \\phi_n^{\\lambda} + \\pot \\xi_n$, with $\\phi_n^{\\lambda} \\in H^1_{\\mathrm{f}}(\\bR^{3N})$, $\\xi_n \\in H^{1\/2}_{\\mathrm{f}}(\\bR^{3N-3})$, and \n\\beq\n\\form^{\\lambda}[\\psi_n - \\psi_m]= \\mathcal{F}_0[\\phi_n^{\\lambda} - \\phi_m^{\\lambda}] + N \\qform[\\xi_n -\\xi_m]\\,.\n\\eeq\nThis, together with \\eqref{cC}, implies that $\\{\\phi_n^{\\lambda} \\}$ is a Cauchy sequence in $ H^1_{\\mathrm{f}}(\\bR^{3N})$ and $\\{ \\xi_n\\}$ is a Cauchy sequence in $H^{1\/2}_{\\mathrm{f}}(\\bR^{3N-3})$. Let us denote by $\\phi^{\\lambda}$ and $\\xi$ the corresponding limits. From the explicit expression of the potential \\eqref{pot xi} we notice that \n\\beq\n\\| \\pot \\xi \\|_{L^2_{\\mathrm{f}}} \\leq c \\, \\|\\xi\\|_{L^2_{\\mathrm{f}}}\\,,\n\\eeq\nwhere $c>0$. Hence,\n\\beq\n\\lim_{n \\rightarrow \\infty} \\|\\psi_n - (\\phi^{\\lambda} + \\pot \\xi) \\|_{L^2_{\\mathrm{f}}}= \\lim_{n \\rightarrow \\infty}  \\| (\\phi_n^{\\lambda} - \\phi^{\\lambda}) + \\pot ( \\xi_n - \\xi) \\|_{L^2_{\\mathrm{f}}}       =0\\,.\n\\eeq\nSince the limit of the $\\psi_n$'s is unique, $\\psi = \\phi^{\\lambda} + \\pot \\xi$. Therefore $\\psi \\in \\dom(\\form)$ and $\\lim_{n \\rightarrow \\infty} \\form^{\\lambda}[\\psi-\\psi_n]=0$. This shows that the form $\\form^{\\lambda}$ is closed and a fortiori $\\form$ is.\n\\end{proof}\n\n\n\n\n \n\n\n\n\n\\section{Unboundedness from Below of $ \\form $} \n\\label{instability: sec}\n\t\t\nThis section is devoted to the proof of Theorem \\ref{ub1}.\nAs we shall see, what makes an instability condition hard to prove is the restriction to antisymmetric wave functions. \n\n\nIn fact, the proof relies on the explicit evaluation of the charge form $ \\qform $ on a trial function, i.e., a convenient sequence of charges with energy going to $ -\\infty $. Identifying one such sequence is easy when $ N = 2 $ because $ \\qform $ is in practice the same as the reduced form $ F_{1} $ -- see \\eqref{michelebuliccio} below -- \nand the analysis performed in Section \\ref{stability: sec} suggests that a convenient $ Q_n(\\pv) $ has to be chosen\nin the subspace with angular momentum $ \\ell=1 $ and such that in the position representation it becomes peaked at the origin as $ n \\to \\infty $ (i.e., two identical fermions coming arbitrarily close).\n\nWhen $ N>2 $, on the other hand, a natural trial function satisfying the antisymmetry constraint would be the Slater determinant of $N$ one-particle charges, one of which is $ Q_n $ itself. A convenient choice is driven by the physical idea of a $N$-body configuration that contains precisely the $(2\\!+\\!1)$-body structure minimizing the energy with $ N = 2 $, whereas all remaining particles are placed far away in space so that there is no or negligible interference with the two-body state. This results in a $N$-particle Slater determinant  between $ Q_n $ and $N-1$ copy of a different component (see \\eqref{min sequence} below). The fermionic character of the trial function is thus fulfilled by construction and optimising the choice of the second component produces only higher order symmetry correlations.\n\n\tThroughout this section we assume that $ \\la $ is a positive number such that $ C_1 \\leq \\la^{-1} \\leq C_2 $ for two finite constants $ C_1, C_2 < \\infty $, which in particular will allow us to incorporate error factors proportional to $ \\la^{-1} $ into a constant $ C $.\n \t\n\t\\begin{proof}[Proof of Theorem \\ref{ub1}]\n\t\tIn order to prove instability of the form $ \\form $, it is enough to produce a sequence of normalised charges $ \\xi_n \\in H_{\\mathrm{f}}^{1\/2}(\\R^{3(N-1)}) $ such that\n\t\t\\beq\n\t\t\t\\lim_{n \\to \\infty} \\qform[\\xi_n] = - \\infty,\n\t\t\\eeq\n\t\tsince the sequence of states $ \\pot \\xi_n $ then satisfies\n\t\t\\beq\n\t\t\t\\form[\\pot \\xi_n] = - \\la \\lf\\| \\pot \\xi_n \\ri\\|_{L^2(\\R^{3N})}^2 + N \\qform[\\xi_n] \\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty.\n\t\t\\eeq\n\t\t\\underline{Case $ N = 2 $}. The result was already proved in \\cite{FT}, but we repeat here the argument for we use a slightly different trial function that turns out to be useful in the general case $ N > 2 $. Owing to  \\eqref{qform 1}, \n\t\t\\beq\\label{michelebuliccio}\n\t\t\t\\qform[\\xi] -  \\alpha \\| \\xi \\|_{L^2(\\R^{3})}^2 = \\sqrt{\\la} F_1 \\lf[ Q \\ri]\n\t\t\\eeq\n\t\twhere $ Q(\\pv) = \\la^{3\/4} \\xi(\\sqrt{\\la}\\pv) $ (recall \\eqref{charge Qk}). Then we only need to produce $ Q_n \\in L^2(\\RT) $ such that $ \\lim_{n \\to \\infty} F_1[Q_n] = - \\infty $. Note that no constraint is imposed on the symmetry properties of $ Q_n $. In fact, according to the discussion of Section \\ref{stability: sec} (see  \\eqref{formula0}, \\eqref{formula1} and \\eqref{S10}), we can take each $ Q_n $ in the subspace with angular momentum $ l = 1 $ and such that the support of its $\\sharp-$transform defined in \\eqref{formula2} gets concentrated at the origin. Explicitly, we choose\n\t\t\\beq\n\t\t\t\\label{qnga}\n\t\t\t\\qnga(\\kv) : = n^{-3\/2} \\qga(n^{-1} k) Y_1^{0}(\\vartheta_k),\n\t\t\\eeq\n\t\t\t\t\\beq\n\t\t\t\\label{Qgamma}\n\t\t\t\\qga(p) : =  \\pi^{-1\/4} c_{\\gamma} \\gamma^{1\/2} p^{-1} \\exp \\lf\\{ - \\hbox{$\\frac{1}{8\\gamma^{2} }$} \\ri\\} \\exp \\lf\\{ - \\half \\gamma^2 \\lf( \\log p \\ri)^2 \\ri\\} \\Theta(p-1),\n\t\t\\eeq\n\t\twhere $ \\gamma \\in (0, 1) $ is a variational parameter, $ \\Theta $ is the Heaviside function, i.e., $ \\Theta(p) = 1 $ if $ p \\geq0 $ and $ 0 $ otherwise, and $ c_{\\gamma} $ is a normalisation constant.\n\t\tBy direct computation,\n\t\t\\beq\n\t\t\t\\lf\\| \\qnga \\ri\\|^2_{L^2} = \\int_1^{\\infty} \\diff p \\: p^2 \\lf| \\qga(p) \\ri|^2 = \\tx\\frac{c^2_{\\gamma}}{\\sqrt{\\pi}} \\disp\\int_{0}^{\\infty} \\diff t \\: \\exp\\lf\\{ - t^2 + \\tx\\frac{t}{\\gamma} - \\tx\\frac{1}{4\\gamma^2} \\ri\\} = \\half c^2_{\\gamma} \\: \\lf( 1 + \\mathrm{erf}\\lf\\{ \\tx\\frac{1}{2\\gamma} \\ri\\} \\ri).\n\t\t\\eeq\n\t\tImposing $\\qnga$ to be normalised yields\n\t\t\\beq\n\t\t\t\\label{cgamma}\n\t\t\t1 \\leq c^2_{\\gamma} = 2 \\lf[ 1 + \\mathrm{erf}\\lf\\{ \\tx\\frac{1}{2\\gamma} \\ri\\} \\ri]^{-1} \\leq 1 + C\\gamma \\exp\\lf\\{-\\tx\\frac{1}{4\\gamma^2} \\ri\\}.\n\t\t\\eeq \n(see, e.g., \\cite[Eq. (7.1.13)]{AS}). It is also useful to compute the following integral for  $ a \\geq -1 $:\n\t\t\\bml{\n\t\t\t\\label{qga est}\n\t\t\t\\int_0^{\\infty} \\diff p \\: p^{2+a} \\lf| \\qga(p) \\ri|^2 = \\tx{\\frac{1}{\\sqrt{\\pi}}} \\gamma c_{\\gamma}^2 \\exp \\lf\\{ -\\tx{\\frac{1}{4\\gamma^2}} \\ri\\} \\disp\\int_{0}^{\\infty} \\diff t \\: \\exp \\lf\\{ - \\gamma^2 t^2 + (1+a) t \\ri\\}\t\\\\\n\t\t\t =\\half c_{\\gamma}^2 \\lf( 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\} \\ri) \\exp \\lf\\{ \\tx{\\frac{1}{4\\gamma^2}} (2a+a^2) \\ri\\} \\leq  \\lf( 1 + C \\gamma \\ri) \\exp \\lf\\{ \\tx{\\frac{1}{4\\gamma^2}} (2a+a^2) \\ri\\},\t\n\t\t}\n\twhere we used\n\t\t\\bdm\n\t\t\t\\half c_{\\gamma}^2 \\lf( 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\} \\ri) = \\frac{ 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1+a}{2\\gamma} \\ri\\}}{ 1 + \\mathrm{erf} \\lf\\{\\tx\\frac{1}{2\\gamma} \\ri\\}} \\leq 1 + \\frac{2}{\\sqrt{\\pi}} \\int_{\\frac{1}{2\\gamma}}^{\\frac{1+a}{2\\gamma}} \\diff t \\: e^{-t^2} \\leq 1 + C \\gamma^{-1} \\exp\\lf\\{- \\tx\\frac{1}{4 \\gamma^2} \\ri\\} \\leq 1 + \\OO(\\gamma).\n\t\t\\edm\n\t\tUsing the decomposition \\eqref{Fzl}, as well as the scaling law of $ \\qnga $ with $n$, we have\n\t\t\\beq\n\t\t\t\\label{est F1 1}\n\t\t\tF_1[\\qnga] = G_{1}^{\\mathrm{diag}}\\lf[n^{-3\/2} \\qga(n^{-1} p)\\ri] + G_{1,1}^{\\mathrm{off}}\\lf[n^{-3\/2} \\qga(n^{-1} p)\\ri] = n \\lf[ G_{n^{-2}}^{\\mathrm{diag}}\\lf[\\qga\\ri] + G_{n^{-2},1}^{\\mathrm{off}}\\lf[\\qga\\ri] \\ri].\n\t\t\\eeq\nWe estimate the diagonal term in \\eqref{est F1 1} as\n\\begin{equation}\\label{approx Goff}\n\\begin{split}\nG_{n^{-2}}^{\\mathrm{diag}}\\lf[\\qga\\ri]\\;&\\leq\\;2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\lf(1 + \\OO(n^{-1}) \\ri) \\big\\| Q_{\\gamma}^{\\sharp} \\big\\|_{L^2}^2\t\\\\\n&\\leq\\;2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf[1 + \\OO(n^{-1}) + \\OO(\\gamma) \\ri],\n\\end{split}\n\\end{equation}\nwhere we used\n\t\t\\beq\n\t\t\t\\label{tilde Q norm}\n\t\t\t\\big\\| Q^{\\sharp}_{\\gamma} \\big\\|_{L^2}^2 = \\int_{\\R} \\diff k \\: e^{4k} \\: \\big| \\hat{Q}_{\\gamma}\\big(e^k\\big) \\big|^2 = \\int_{0}^{\\infty} \\diff p \\: p^3 \\: | Q_{\\gamma}(p) |^2 \\leq  (1 + C \\gamma ) \\exp \\lf\\{ \\tx{\\frac{3}{4\\gamma^2}} \\ri\\}.\n\t\t\\eeq\n\t\tAs for the off-diagonal term in \\eqref{est F1 1},\n\\beq\\label{Goff1}\nG^{\\mathrm{off}}_{n^{-2},1} \\lf[ \\qga \\ri] =   G^{\\mathrm{off}}_{0,1} \\lf[ \\qga \\ri] + \\mathcal{R},\n\\eeq\nwhere\n\\bml{\n \t\t\t|\\mathcal{R}| \\leq C \\int_{\\R^3} \\diff \\sv \\int_{\\R^3} \\diff \\tv \\: \\bigg| \\lf(s^2 + t^2 + \\tx\\frac{2}{m+1} \\sv \\cdot \\tv + n^{-2}\\ri)^{-1} - \\lf(s^2 + t^2 + \\tx\\frac{2}{m+1} \\sv \\cdot \\tv\\ri)^{-1} \\bigg| \\lf| \\qga(s) \\ri| \\lf| \\qga(t) \\ri|\t\\\\\n\t\t\t\\leq C n^{-2} \\int_{\\R^3} \\diff \\sv \\int_{\\R^3} \\diff \\tv \\: \\frac{\\lf| \\qga(s) \\ri| \\lf| \\qga(t) \\ri|}{(s^2 + t^2)^{2}}  \\leq C n^{-2} \\int_{\\R^3} \\diff \\sv \\: \\lf| \\qga(s) \\ri|^2 \\int_{1}^{\\infty} \\diff t \\: t^{-2} \\leq \\OO(n^{-2}). \\label{R1}\n\t\t}\nMoreover,\n\\begin{equation}\\label{Goff est}\n\\begin{split}\nG^{\\mathrm{off}}_{0,1} \\lf[ \\qga \\ri]\\;&=\\;\\int_{\\R} \\diff k \\: S_1(k) \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2  \\\\\n& \\leq\\; S_1(0) \\big\\| Q^{\\sharp}_{\\gamma} \\big\\|_{L^2}^2 + \\int_{\\R} \\diff k \\: \\lf( S_1(k) - S_1(0) \\ri) \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2 \\\\\n&\\leq\\; S_1(0)  \\exp \\lf\\{ \\tx{\\frac{3}{4\\gamma^2}} \\ri\\} \\lf(1 + \\OO(\\gamma)  \\ri) + C \\int_{\\R} \\diff k \\: \\sqrt{|k|} \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2 \\\\\n&=\\; - 2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} \\Lambda(m,2) \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf( 1 + \\OO(\\gamma)  \\ri) + C \\int_{\\R} \\diff k \\: \\sqrt{|k|} \\big| Q^{\\sharp}_{\\gamma}(k) \\big|^2\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{S10} and the elementary estimate $S_1(k)-S_1(0) \\leq C \\sqrt{|k|}$. To estimate the last integral in \\eqref{Goff est} we observe that\n\\begin{equation}\\label{Qshes}\n\\begin{split}\nQ^{\\sharp}_{\\gamma}(k)\\;&=\\;\\pi^{-1\/4} c_{\\gamma} \\gamma^{1\/2}  \\exp\\lf\\{\\tx-\\f{1}{8 \\gamma^2} \\ri\\}    \\f{1}{\\sqrt{2\\pi}} \\int_0^{\\infty} \\!\\!\\!dx\\, \n\\exp\\lf\\{\\tx-ikx -\\f{\\gamma^2}{2} x^2 +x\\ri\\}  \\\\\n&=\\;\\pi^{-1\/4} c_{\\gamma} \\gamma^{-1\/2} \n\\exp\\lf\\{\\tx-\\f{1}{8 \\gamma^2}\\ri\\} \\bigg(\n\\exp\\lf\\{\\tx-\\f{k^2}{2\\gamma^2} -i\\f{k}{\\gamma^2} +\\f{1}{2\\gamma^2}\\ri\\} - \\f{1}{2 \\sqrt{\\pi} z}(1+r(z)) \\bigg),\n\\end{split}\n\\end{equation}\nwhere\n\\beq\nz= \\f{1-ik}{\\sqrt{2}\\, \\gamma}\\,, \\hspace{1cm} |r(z)|\\leq \\f{\\gamma^2}{\\sqrt{1+k^2}}\\,.\n\\eeq\nTherefore,\n\\begin{equation}\\label{sqrQ}\n\\begin{split}\n\\!\\!\\!\\!\\!\\!\\int_{\\bR}\\!\\!\\!dk\\, \\sqrt{|k|} |Q^{\\sharp}_{\\gamma} (k)|^2 \\;&\\leq\\;\\f{2 \\, c_{\\gamma}^2}{\\sqrt{\\pi} \\gamma}  \n\\exp\\lf\\{\\tx\\f{3}{4\\gamma^2}\\ri\\} \\int_{\\bR}\\!\\!\\! dk\\, \\sqrt{|k|} \\, \n\\exp\\lf\\{\\tx-\\f{k^2}{\\gamma^2}\\ri\\}  \n+ \\f{4 \\, c_{\\gamma}^2 \\, \\gamma}{\\pi^{3\/2}} \n\\exp\\lf\\{\\tx-\\f{1}{4\\gamma^2} \\ri\\}\n\\int_{\\bR} \\!\\!\\!dk \\, \\f{\\sqrt{|k|}}{1+\\, k^2} \\\\\n& \\leq\\;C \\, \\sqrt{\\gamma} \\; \\exp\\lf\\{\\tx\\f{3}{4 \\gamma^2} \\ri\\} \\Big( 1 + \\sqrt{\\gamma} \\, \n\\exp\\lf\\{\\tx-\\f{1}{\\gamma^2}\\ri\\} \\Big).\n\\end{split}\n\\end{equation}\nUsing \\eqref{approx Goff}, \\eqref{Goff1}, \\eqref{R1}, \\eqref{Goff est}, \\eqref{sqrQ} in  \\eqref{est F1 1} we finally obtain\n\t\t\\beq\n\t\t\tF_1[\\qnga] \\leq 2 \\pi^2 \\frac{\\sqrt{m(m+2)}}{m+1} n \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\lf[ 1 - \\Lambda(m,2) + \\OO(\\sqrt{\\gamma}) + \\OO(n^{-1}) \\ri] \\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty,\n\t\t\\eeq\n\t\tif $ \\Lambda(m,2) > 1 $ and $ \\gamma $ is taken small enough (independent of $ n $).\n\n\\noindent \\underline{Case $N>2$}. \n\t\tAs mentioned at the beginning of this section, this case is more complicated, for the trial sequence $ \\xi_n $ must be antisymmetric under the exchange of any variable, i.e., $\\xi_n \\in \\ldf(\\R^{3N-3}) $, and at the same time we want $ \\hat\\xi_n(\\kv_1,\\ldots,\\kv_{N-1}) $ to behave like $ \\qnga(\\kv_1) $ once the other degrees of freedom are traced out. Looking for $\\xi_n$ matching these two requirements is an example of the well-known representability problem (see, e.g., \\cite{LS}), i.e., the search for sufficient conditions to impose on a one-particle density matrix so that it can be obtained as the reduced  density matrix of a fermionic many-body state. We remark that the solution is known only in some special cases and is non-trivial. Our choice here is a trial state that is as close as possible to an uncorrelated state, which is given by an antisymmetric wave function containing $ \\qnga $. Explicitly,\n\t\t\\beq\n \t\t\t\\label{min sequence}\n\t\t\t\\hat\\xi_n(\\kv_1, \\ldots, \\kv_{N-1}) : = \\frac{1}{\\sqrt{(N-1)!}}\n\t\t\t\\lf|\n\t\t\t\\begin{array}{ccccc}\n\t\t\t\t\\qnga(\\kv_1)\t&\t\\Xi_{\\beta,2}(\\kv_1)\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{1})\t\t\\\\\n\t\t\t\t\\qnga(\\kv_2)\t&\t\\Xi_{\\beta,2}(\\kv_2)\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{2})\t\t\\\\\n\t\t\t\t\\vdots \t&\t\\vdots \t&\t\\mbox{} \t&\t\\vdots\t\t\\\\\n\t\t\t\t\\qnga(\\kv_{N-1})\t&\t\\Xi_{\\beta,2}(\\kv_{N-1})\t&\t\\cdots \t&\t\\Xi_{\\beta,N-1}(\\kv_{N-1})\t\n\t\t\t\\end{array}\n\t\t\t\\ri|\\,,\n\t\t\\eeq\n\t\twhere $ \\qnga $ is defined in \\eqref{qnga}, $ 0 < \\beta \\ll 1 $ is another variational parameter,\n\t\t\\beq\n\t\t\t\\label{chibm}\n\t\t\t\\chibm(\\kv) := (4\\pi)^{-1\/2} \\beta^{-3\/2} \\: \\Xi(\\beta^{-1} k) \\: \\exp\\lf\\{ i l \\varphi_k \\ri\\},\n\t\t\\eeq\n\t\t$ l \\in \\N $, $ \\Xi \\in C^{\\infty}_0(\\R^+) $ is real-valued, with support in $ (0,1) $, and such that\n\t\t\\beq\n\t\t\t\\label{G normalized}\n\t\t\t\\int_{0}^1 \\diff k \\: k^2 \\: \\Xi^2(k) = 1.\n\t\t\\eeq\n\t\tNote that, since the two functions $ \\qnga $ and $ \\chibm $, $ l > 0 $, are orthonormal by construction, the function \\eqref{min sequence} belongs to $ \\ldf(\\R^{3(N-1)}) $ and  is normalised. Moreover, the supports of $ \\qga $ and $ \\Xi $ do not intersect, which implies that the supports of $ \\qnga $ and $ \\chibm $ are disjoint as well, provided $ \\beta \\leq n $, which follows from the assumptions on $ \\beta $. \n\t\t\n\t\tWe can now evaluate $\\dqform[\\xi_n]$. We start by estimating the diagonal part. Using the exchange symmetry and the definition of $ L_{\\la} $ in \\eqref{Lla}),\n\t\t\\bml{\n \t\t\t\\label{energy est 0}\n\t\t\t\\dqform[\\xi_n] = \\al +  \\frac{1}{(N-2)!} \\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\diff \\kkv \\: L_{\\la}(\\kv_1,\\ldots,\\kv_{N-1}) \\lf| \\qnga(\\kv_1) \\ri|^2 \t\\cdot \\\\\n\t\t\t\\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-1}} \\prod_{l,j =2}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j}(\\kv_{\\tau(j)}),\n\t\t\t}\n\t\twhere $ \\mathcal{P}_{N-1} $ is the group of permutations of $ N-2 $ elements $2, \\ldots, N - 1 $ and $ \\sgn(\\sigma) $ denotes the sign of any $ \\sigma \\in \\mathcal{P}_N $. All the other terms vanish because of the integral of the product $ \\qnga(\\kv_i) \\chib(\\kv_i) $, which is pointwise zero thanks to the disjoint supports of the functions. Extracting the main factor\n\t\t\\bdm\n\t\t\t\\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la}\\,,\n\t\t\\edm\n\t\tand bounding the rest by means of the inequality\n\t\t\\beq\n\t\t\t\\label{useful ineq 1}\n\t\t\t\\sqrt{a + b} \\leq \\sqrt{|a|} + \\sqrt{|b|},\t\\hspace{1cm}\t\\mbox{for} \\:\\: a + b \\geq 0\\,,\t\n\t\t\\eeq\n\t\twe obtain \n\\begin{equation}\\label{Lla estimate}\n\\begin{split}\nL_{\\la}(\\kv_1, \\ldots, \\kv_{N-1}) \\;&\\leq\\; 2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\,\\bigg\\{ 1 + \\lf(\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la\\ri)^{-1\/2}  \\times \\\\\n& \\qquad\\qquad\\times\\bigg[ \\tx\\frac{m(m+2)}{(m+1)^2} \\disp\\sum_{i = 2}^{N-1} k_i^2 + \\tx\\frac{2m}{(m+1)^2} \\bigg| \\disp\\sum_{j > 1} \\kv_1 \\cdot \\kv_j +  \\disp\\sum_{1 < i <j} \\kv_i \\cdot \\kv_j \\bigg| \\bigg]^{1\/2} \\bigg\\}\t\\\\\n&\\leq\\;2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\bigg\\{ 1 + C_{N} \\disp\\sum_{i = 2}^{N-1} \\lf( k_i + \\sqrt{k_1k_i} \\ri) \\bigg\\}\\,.\n\\end{split}\n\\end{equation}\nThe diagonal term can be estimated as\t\n\t\t\\bml{\n \t\t\t\\label{energy est 1}\n\t\t\t\\dqform[\\xi_n] - \\al \\leq  2 \\pi^2\\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\:  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\lf| \\qnga(\\kv_1) \\ri|^2 \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2\t\\\\\n\t\t\t+ C_N \\int_{\\R^{3(N-1)}} \\diff \\kv_1 \\cdots \\diff \\kv_{N-1} \\: \\lf|L_{\\la}(\\kv_1, \\kkv) - 2 \\pi^2 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\ri|  \\lf| \\qnga(\\kv_1) \\ri|^2 \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2,\n\t\t}\n\t\tthanks to the orthogonality of functions $ \\Xi_{\\beta,l} $  and  $ \\Xi_{\\beta,l^{\\prime}} $ for $ l \\neq l^{\\prime} $. Thus, by \\eqref{Lla estimate}, $ \\dqform[\\xi_n] - \\al $ is bounded from above by\n\\begin{equation}\n\\begin{split}\n\\!\\!\\!\\!\\!\\!\\!2 \\pi^2\\!\\!\\!& \\int_{\\R^3} \\diff \\kv_1 \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\la} \\lf| \\qnga(\\kv_1) \\ri|^2 \\int_{\\R^{3(N-2)}} \\diff \\kkv \\bigg[ 1 + C_N \\bigg( k_2 + \\sqrt{k_1 k_2} \\bigg) \\bigg] \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2 \\\\\n&\\leq\\;2 \\pi^2 n \\int_{\\R^3} \\diff \\kv_1  \\sqrt{\\tx\\frac{m(m+2)}{(m+1)^2} k_1^2 + \\frac{\\la}{n^2}}  \\lf| \\qga(\\kv_1) \\ri|^2 \\int_{0}^{1} \\diff k_2 \\: k_2^2\\lf[ 1 + C_N \\lf( \\beta k_2 + \\sqrt{n \\beta k_1 k_2} \\ri) \\ri] \\lf| \\Xi(k_2) \\ri|^2 \\\\\n& \\leq 2 \\pi^2 n \\tx\\frac{\\sqrt{m(m+2)}}{m+1} \\lf(1 + \\OO(n^{-1}) \\ri) \\disp\\int_{1}^{\\infty} \\diff k_1 \\: k_1^3 \\lf[ 1 + C_N \\lf( \\beta + \\sqrt{n \\beta k_1} \\ri) \\ri] \\lf| \\qga(k_1) \\ri|^2 \\\\\n& \\leq\\;2 \\pi^2 n \\tx\\frac{\\sqrt{m(m+2)}}{m+1}  \\exp \\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\}  \\lf[1 + C _N \\lf( \\sqrt{n \\beta} \\exp \\lf\\{ \\tx{\\frac{9}{16 \\gamma^2}} \\ri\\} + \\gamma +  \\beta + n^{-1} \\ri) \\ri]\n\\end{split}\n\\end{equation}\nwhere we used \\eqref{qga est}. We now compute the off-diagonal term (recall \\eqref{oqform}).  Owing to the exchange symmetry, the pre-factor $ N-1 $ cancels with the normalisation factor of $ \\xi_n $ and\n\t\t\\bml{\n \t\t\t\\label{energy est 3}\n\t\t\t\\oqform[\\xi_n] = \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kkv \\:\t\\green(\\sv, \\tv, \\kkv) \\bigg\\{ \\qnga^*(\\sv) \\qnga(\\tv) \\times \t\\\\\n\t\t\t\\times\\frac{1}{(N-2)!}\\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-1}} \\prod_{l,j =2}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j}(\\kv_{\\tau(j)}) \\bigg\\}  + \\mathcal{R}\n\t\t\n\t\t\n\t\t \n\t\t}\n\t\twhere $ \\mathcal{R} $ contains some remainder terms. We estimate the leading term (first term on the r.h.s.~of \\eqref{energy est 3}) from above by\n\t\t\\bml{\n \t\t\t\\label{energy est 4}\n\t\t\tn \\int_{\\R^{6}} \\diff \\sv \\diff \\tv \\:  G_{\\la\/n^2}(\\sv, \\tv) \\qga(s) \\qga(t) Y_1^0(\\vartheta_s) Y_1^0(\\vartheta_t)\t\\\\\n\t\t\t + C_N \\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kkv \\:\t\\green(\\sv, \\tv, \\kkv) \\green(\\sv, \\tv) \\lf[ k_2^2 + (s+t) k_2 \\ri] \t\\lf| \\qnga(\\sv) \\ri| \\lf| \\qnga(\\tv) \\ri| \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2,\t\t\t}\n\t\twhere in the first term we replaced $ \\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) $ with $ \\green(\\sv, \\tv, \\zerov, \\ldots, \\zerov) = :  \\green(\\sv, \\tv) $ and we exploited the orthogonality of functions $ \\Xi_{\\beta,l} $  and  $ \\Xi_{\\beta, l^{\\prime}} $ for $ l \\neq l^{\\prime} $.\n\t\tThe first term in the expression above was bounded  in \\eqref{Goff est}.  Using\n\t\t\\beq\n\t\t\t\\label{kernel est 1}\n\t\t\t\\green(\\sv,\\tv,\\kv_2, \\ldots, \\kv_{N-1}) \\leq \\bigg[ \\frac{m}{m+1} \\big(s^2 + t^2\n\t\t\t\\big) + \\la \\bigg]^{-1}\n\t\t\\eeq \n\tand the elementary inequality $x+y \\leq \\sqrt{x^2 +1} \\sqrt{y^2 +1}$, $x,y \\geq 0$, \tthe second term in \\eqref{energy est 4} can be estimated as\n\\begin{equation}\\label{off2}\n\\begin{split}\nC_N \\int_{\\R^{3N}}& \\diff \\sv \\diff \\tv \\diff \\kkv   \\f{k_2^2 +(s+t) k_2}{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} \\big]^2}  \\lf| \\qnga(\\sv) \\ri| \\lf| \\qnga(\\tv) \\ri| \\prod_{l =2 }^{N-1} \\lf| \\Xi_{\\beta,0}(\\kv_l)\\ri|^2 \\\\\n&\\leq\\;C_N \\, n^{-1}  \\int_{\\R^{6}} \\!\\! \\diff \\sv \\diff \\tv  \\f{\\beta^2 +n \\beta (s+t) }{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} n^{-2} \\big]^2}  \\lf| \\qga(\\sv) \\ri| \\lf| \\qga(\\tv) \\ri|\\\\\n&\\leq\\; C_N n^{-1} (\\beta^2 + n \\beta)  \\bigg[ \\sup_{\\tv} \\int_{\\R^{3}} \\!\\! \\diff \\sv   \\f{1 }{\\big[ s^2 +t^2 + \\lambda \\f{m+1}{m} n^{-2} \\big]^2} \\bigg] \\int_{\\R^3} \\!\\! \\diff \\pv \\, (p^2 +1) |Q_{\\gamma}(\\pv)|^2 \\\\\n&\\leq C_N (\\beta^2 + n \\beta) (1+\\gamma) \\exp \\lf\\{ \\tx\\frac{2}{\\gamma^2} \\ri\\}\\,.\n\\end{split}\n\\end{equation}\nThe rest $ \\mathcal{R} $ in \\eqref{energy est 3} contains several terms but it is not difficult to see that most of them vanish because of the disjoint supports of $ \\qnga $ and $ \\chib $. What remains is\n\\begin{equation}\\label{remainder 1}\n\\begin{split}\n[(N-3)!]^{-1} &\\int_{\\R^{3N}} \\diff \\sv \\diff \\tv \\diff \\kv_2 \\cdots \\diff \\kv_{N-1} \\:\t\\green(\\sv, \\tv, \\kv_2, \\ldots, \\kv_{N-1}) \\: \\Xi_{\\beta,1}^*(\\sv) \\Xi_{\\beta,1}(\\tv)  \\lf| \\qnga(\\kv_2) \\ri|^2 \\times \\\\\n& \\qquad\\qquad\\times \\sum_{\\sigma,\\tau \\in \\mathcal{P}_{N-2}} \\prod_{l,j = 3}^{N-1} \\sgn(\\sigma) \\sgn(\\tau) \\Xi_{\\beta,l}^*(\\kv_{\\sigma(l)}) \\Xi_{\\beta,j} (\\kv_{\\tau(j)}) \\\\ \n&\\leq\\;\tC_N \\int_{\\R^{6}} \\diff \\sv \\diff \\tv \\:  \\lf| \\Xi_{\\beta,1}(\\sv)  \\ri| \\lf| \\Xi_{\\beta,1}(\\tv) \\ri| \\leq C_N \\beta^3,\n\\end{split}\n\\end{equation}\nwhere we exploited the exchange symmetry again, as well as the properties of $ \\Xi $ (in particular $ \\supp(\\Xi) \\subset (0,1) $) and \\eqref{kernel est 1}. It is understood that the sum over permutations as well as the following factor in \\eqref{remainder 1} is not present when $N = 3 $. Putting together \\eqref{energy est 3}, \\eqref{energy est 4}, \\eqref{off2} and \\eqref{remainder 1}, we obtain \n\t\t\\beq\n \t\t\t\\label{energy est 8}\n\t\t\t\\oqform[\\xi_n] \\leq n G^{\\mathrm{off}}_{\\la\/n^{2},1} \\lf[ \\qga \\ri] + C_N \\lf(\\beta^2 + n \\beta \\ri) \\exp \\lf\\{ \\tx\\frac{2}{\\gamma^2} \\ri\\},\n\t\t\\eeq\n\t\tand finally\n\t\t\\bml{\n \t\t\t\\label{energy est final}\n\t\t\t\\qform[\\xi_n] \\leq 2 \\pi^2 n \\frac{\\sqrt{m(m+2)}}{m+1} \\exp\\lf\\{ \\tx\\frac{3}{4\\gamma^2} \\ri\\} \\Big\\{ 1 -  \\Lambda(m,2) \t\\\\\n\t\t\t   + C_N \\lf[ \\al \\,  n^{-1} + \\sqrt{\\gamma} + \\sqrt{n \\beta} \\exp \\lf\\{ \\tx{\\frac{9}{16 \\gamma^2}} \\ri\\} + (n^{-1} \\beta^2 +\\beta) \\exp\\lf\\{ \\tx\\frac{5}{4\\gamma^2} \\ri\\}  \\ri] \\Big\\}\\,.\n\t\t}\n\t\tBy assumption, $  1 -  \\Lambda(m,2) < 0 $ and we choose $ \\beta \\ll n^{-1} $ as, say, $ \\beta = n^{-2} $. Thus, we can always find some small $ \\gamma = \\OO(1) > 0 $, such that \n\t\t\\beq\n\t\t\t\\qform[\\xi_n]\n\t\t\t\\xrightarrow[\\;n\\to\\infty\\;]{} - \\infty\n\t\t\\eeq\nwhich concludes the proof.\n\\end{proof}\n\n\n\n\\vspace{1cm}\n\\n\t\t\n{\\bf Acknowledgments}. M.C. acknowledges the support of the European Research Council under the European Community Seventh Framework Program (FP7\/2007-2013 Grant Agreement CoMBos No. 239694).\n\n\n\n\\vs\n\n\\section*{Appendix}\n\n\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\setcounter{subsection}{0}\n\\setcounter{pro}{0}\n\\renewcommand{\\thesection}{A}\n\n\n\\label{Appendix A}\n\nHere we describe the formal procedure for the construction of the quadratic form $\\form$.   \nWe start  from the  Hamiltonian \\eqref{cm Hamiltonian} written in the Fourier space\n\\bml{\n(\\widehat{H \\psi})(\\kv_1,\\ldots, \\kv_N) = \\bigg(\\sum_{i=1}^N k_i^2 +\\f{2}{m\\!+\\!1}\\sum_{i<j} \\kv_i \\cdot \\kv_j \\bigg) \\hat{\\psi}(\\kv_1,\\ldots, \\kv_N) \\\\\n  +\\f{\\mu}{(2 \\pi)^3} \\int_{\\bR^3} \\!\\! \\diff \\sv \\,\\hat{\\psi} (\\kv_1,\\ldots, \\kv_{i-1},\\sv,\\kv_{i+1},\\ldots,\\kv_N)\n}\nand we consider the corresponding quadratic form, regularized by means of an ultra-violet cut-off\n\\bml{\n\t\\label{ren Form}\n\t\\renform[\\psi] : = \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\bigg\\{ \\sum_{i =1}^N k_i^2 + \\f{2}{(m+1)} \\sum_{i < j} \\kv_i \\cdot \\kv_j \\bigg\\} \\lf| \\hat{\\psi}(\\kv_1, \\ldots, \\kv_N) \\ri|^2\t\\\\\n\t + \\frac{\\mu(\\alpha,R)}{(2\\pi)^{3}} \\sum_{i = 1}^N   \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i)  \\hat{\\psi}^*(\\kv_1,\\ldots, \\kv_N) \\times \t\\\\\n\t\\times\\int_{\\RT} \\diff \\sv \\: \\chi_R(s) \\hat{\\psi}(\\kv_1,\\ldots, \\kv_{i-1},\\sv,\\kv_{i+1},\\ldots,\\kv_N)\\,.\n}\nHere $ \\psi \\in H^1(\\R^{3N}) $, $ \\chi_R(s) $ is the characteristic function of the three-dimensional ball $ s \\leq R $, and $ \\alpha $ is a  parameter that has the role of a renormalised coupling constant.  Note also that we introduced in $ \\mu$ a dependence on $ R $: the choice of such an explicit dependence will be the main content of the renormalisation procedure.\n\nWe now define the ``surface charges'' $ \\xi_{i}^R \\in L^2(\\R^{3N-3}) $ associated with $ \\psi \\in H^1(\\R^{3N}) $ as\n\\beq\n\t\\label{ren charges}\n\t\\hat{\\xi}_i^R(\\kv_1,\\ldots,\\kv_{N-1}) : = \\frac{\\mu(\\alpha,R)}{(2\\pi)^3} \\int_{\\RT} \\diff \\sv \\: \\chi_R(s) \\hat{\\psi}(\\kv_1,\\ldots,\\kv_{i-1},\\sv,\\kv_{i}, \\ldots, \\kv_{N-1}),\n\\eeq\nand the corresponding ``volume charges'' $\\hat{\\rho}_i^R (\\kv_1,\\ldots,\\kv_N):= \\chi_R(k_i)  \\: \\hat{\\xi}_i^R(\\breve \\kkv_i)$. Further, we introduce    the  ``potential'' \n\\beq\n\t\\label{ren potential}\n\t\\widehat{\\pot  \\rho^R}(\\kv_1, \\ldots, \\kv_N) : = \\sum_{i=1}^N \\green(\\kv_1,\\ldots,\\kv_N) \\: \\chi_R(k_i)  \\: \\hat{\\xi}_i^R(\\breve \\kkv_i),\n\\eeq\nwhere $G_{\\la}$ is defined in \\eqref{Green function} for any $ \\la > 0 $.\nSetting\n\\beq\n\t\\label{ren regular part}\n\t\\hat{\\phi}_{\\la}^R : = \\hat{\\psi} - \\widehat{\\pot \\rho^R},\n\\eeq\nwe have\n\\beq\n\t\\label{ren form decomposition}\n \t\\renform[\\psi] = \\F_0\\lf[\\phi_{\\la}^R\\ri] + \\la \\lf\\| \\phi_{\\la}^R \\ri\\|_{L^2(\\R^{3N})}^2 - \\la \\lf\\| \\psi \\ri\\|_{L^2(\\R^{3N})}^2 + \\renqform\\lf[\\xi^R\\ri],\n\\eeq\nwith $ \\F_0[\\phi] : = \\bra{\\phi} H_0 \\ket{\\phi} $, and\n\\bml{\n\t\\label{ren qform}\n\t\\renqform\\lf[\\xi\\ri] : = - \\sum_{i=1}^N \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i) \\: \\hat\\xi_i^*(\\breve \\kkv_i) \\lf[ \\hat{\\psi}(\\kv_1, \\ldots, \\kv_N) +  \\green(\\kv_1, \\ldots, \\kv_N) \\: \\hat\\xi_i(\\breve \\kkv_i) \\ri]\t\\\\\n\t- \\sum_{i < j} \\int_{\\R^{3N}} \\diff \\kv_1 \\cdots \\diff \\kv_N \\: \\chi_R(k_i) \\:\t\\hat\\xi_i^*(\\breve \\kkv_i) \\green(\\kv_1, \\ldots, \\kv_N) \\: \\chi_R(k_j) \\: \\hat\\xi_j(\\breve \\kkv_j)\\,.\n}\nIn the limit $ R \\to \\infty $ we assume that $\\rho^R_i , \\, \\xi^R_i \\rightarrow \\xi_i$. Moreover, we extract from the diagonal part of \\eqref{ren qform} only the terms not vanishing in that limit\n\\bmln{\n\t\\sum_{i=1}^N \\int_{\\R^{3N-3}} \\diff \\breve \\kkv_i \\: \\lf| \\hat\\xi_i (\\breve \\kkv_i) \\ri|^2 \\lf[ - \\frac{(2\\pi)^3}{\\mu(\\al,R)} - \\int_{\\RT} \\diff \\kv_i \\: \\chi_R(k_i) \\green(\\kv_1, \\ldots, \\kv_N) \\ri] \t\\\\\n\t= \\sum_{i=1}^N \\int_{\\R^{3N}}  \\diff \\breve \\kkv_i \\: \\lf| \\hat\\xi_i (\\breve \\kkv_i) \\ri|^2 \\bigg[ - \\frac{(2\\pi)^3}{\\mu(\\al,R)} - 4 \\pi R \\\\\t\n\t+ 2 \\pi^2 \\bigg[ \\frac{m(m+2)}{(m+1)^2} \\sum_{j \\neq i} k_j^2 + \\frac{2m}{(m+1)^2} \\sum_{i \\neq j} \\kv_i \\cdot \\kv_j + \\la \\bigg]^{1\/2}  + o(1) \\bigg].\n}\nIn order to remove the cut-off one is thus forced to set $ \\mu \\to 0 $ as $ R \\to \\infty $ and, although several choices are allowed, we set\n\\beq\n\t\\mu(\\al,R) : = - \\frac{(2\\pi)^3}{4 \\pi R + \\al},\n\\eeq\nthis way canceling the singular term proportional to $- 4\\pi R $ contained in the expression above. \n\nWe can now remove the cut-off taking the limit $ R \\to \\infty $ and so recovering the expression \\eqref{form}. Note that we exploit at this stage the fermionic symmetry, which in particular implies that all charges can be expressed in terms of a single function $ \\xi $, i.e., \n\\beq\n\t\\xi_i(\\xv_1,\\ldots,\\xv_{N-1}) = (-1)^{i+1} \\xi(\\xv_1,\\ldots,\\xv_{N-1}),\n\\eeq\nand $ \\xi $ itself is totally antisymmetric under exchange of coordinates. This in turns implies that the sign in front of the off-diagonal term is the opposite than in the bosonic case, implying a completely different behavior of the ground state. \n\n\n\n\n\\vs\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith the fast-paced advancement of the Artificial Intelligence (AI) and Robotics fields, there is an increasing potential to resort to\\textit{ robot assistants} (or \\textit{service robots}) to help with daily tasks. Service robots can take on many roles. They can operate as patient carers \\cite{bajones_hobbit_2018}, door-to-door garbage collectors \\cite{ferri_dustcart_2011}, Health and Safety monitors \\cite{dong_design_2018}, museum or tour guides \\cite{waldhart_reasoning_2019}, to name a few. Yet, even decades after the first robot vacuum cleaner was deployed (\\url{https:\/\/en.wikipedia.org\/wiki\/Robotic_vacuum_cleaner}), robot assistants still perform unreliably on more complex tasks. Succeeding in the real world is indeed a difficult challenge because it requires robots to make sense of the high-volume and diverse data coming through their perceptual sensors. Although different sensory modalities contribute to the robot's \\textit{sensemaking} abilities (e.g., touch, sound, temperature), in this work, we focus on the modality of vision. From this entry point, the problem then becomes one of enabling robots to correctly interpret the stimuli of their vision system, with the support of background knowledge sources, a capability also known as \\textit{Visual Intelligence} \\cite{chiatti_towards_2020}. The first prerequisite to Visual Intelligence is the ability to robustly recognise the different objects occupying the robot's environment (\\textit{object recognition}). Let us consider the case of HanS, the Health and Safety robot inspector currently under development at the Knowledge Media Institute (KMi). HanS is expected to monitor the Lab space in search of potentially dangerous situations, such as fire hazards. For instance, imagine that Hans was observing a flammable object (e.g., a paper cup) left on top of a portable heater. To conclude that it is in the presence of a potential fire hazard, the robot would first need to recognise that the cup and the portable heater are there. \n\nCurrently, the most common approach to tackling object recognition tasks is applying methods which are based on Machine Learning (ML). In particular, the state-of-the-art performance is defined by the latest approaches based on Deep Learning (DL) \\cite{liu_deep_2020,lecun_deep_2015}. Despite their popularity, these methods have received many critiques due to their brittleness and lack of transparency \\cite{marcus2018deep,parisi_continual_2019,pearl_theoretical_2018}. To compensate for these limitations, a more recent trend among AI researchers has been to combine ML with knowledge-based reasoning, thus adopting a \\textit{hybrid approach} \\cite{aditya_integrating_2019,gouidis_review_2020}. A question remains, however, on what type of knowledge resources and reasoning capabilities should be leveraged within this new class of hybrid methods \\cite{daruna2018sirok}.\n\nIn \\cite{chiatti_towards_2020}, we identified a set of \\textit{epistemic requirements}, i.e., a set of capabilities and knowledge properties, required for service robots to exhibit Visual Intelligence. We then mapped the identified requirements to the types of classification errors emerging from one of HanS' scouting rounds, where we relied solely on Machine Learning to recognise the objects. This error analysis highlighted that, in 74\\% of the cases, a more accurate object classification could in principle have been achieved if the relative size of objects was considered for their categorisation. For instance, back to HanS' case, the paper cup could be mistaken for a rubbish bin, due to its shape. However, rubbish bins are typically larger than cups. With this awareness, HanS would be able to rule out object categories, which, albeit being visual similarity to the correct class, are \\textit{implausible} from the standpoint of size.\n\nThese elements of \\textit{typicality} and \\textit{plausible reasoning} \\cite{davis_commonsense_2015} link size reasoning to the broader AI objective of developing systems which exhibit \\textit{common sense} \\cite{levesque_common_nodate}, especially with respect to understanding a set of intuitive physics rules governing the environment \\cite{hayes_second_1988,lake_building_2017}. This view is also supported by studies of human visual cognition, which suggest that our priors about the canonical size of objects play a role in how we categorise, draw and imagine objects \\cite{rosch_principles_1999,hoffman_visual_2000,konkle_canonical_2011}.\n\nOn a more practical level, knowledge representations which encode object sizes have already been applied effectively to Natural Language Processing (NLP) tasks. These include answering questions, such as, \"is object A larger than object B?\" \\cite{bagherinezhad_are_2016,elazar_how_2019}. However, despite this body of theoretical and empirical evidence, the role of size in object recognition has received little attention in the field of Computer Vision. To address this issue, in this paper we investigate the performance effects of augmenting a ML-based object recognition system both with background knowledge about the typical size of objects, as well as with a method to reason about the size of the observed objects. Namely, we propose: \n\n\\begin{itemize}\n\\item A hybrid method to validate ML-based predictions based on the typical size of objects. \n\\item A novel representation for size, which categorises objects differently along different dimensions contributing to their size (i.e., their front surface area, depth and aspect ratio). This representation also allows us to model object categories that include instances of varying size.\n\\end{itemize}\n\n\\section{Related work}\n\nState-of-the-art object recognition methods rely heavily on Machine Learning, as further discussed in Section 2.1. Because of the limitations of ML-based methods, hybrid methods, which combine ML with background knowledge and knowledge-based reasoning, have been recently proposed (Section 2.2). In particular, our hypothesis is that awareness of object size has the potential to drastically improve the performance of hybrid object recognition methods \\cite{chiatti_towards_2020}. Therefore, we will conclude our review of the literature by discussing existing approaches to representing the size of objects (Section 2.3). \n\n\\subsection{Machine Learning for Object Recognition}\nThe impressive performance exhibited by object recognition methods based on DL has led to significant advances on several Computer Vision benchmarks \\cite{liu_deep_2020,krizhevsky_imagenet_2017,he_deep_2016}. Deep Neural Networks (NNs), however, come with their own limitations. These models (i) are notoriously data-hungry, i.e., require thousands of annotated training examples to learn from, (ii) learn classification tasks offline, i.e., assume to operate in a closed world, on a fixed and pre-determined set of objects \\cite{mancini_knowledge_2019}, and (iii) learn representational patterns automatically, by iterating over a raw input set \\cite{lecun_deep_2015}. The latter trait can drastically reduce the start-up costs of feature engineering. However, it also complicates tasks such as explaining the obtained features and augmenting them with explicit knowledge statements \\cite{marcus2018deep,pearl_theoretical_2018}.\n\nThe issue of learning robust object representations to adequately reflect changes in the environment, even from minimal training examples, has inspired the development of few-shot metric learning methods. \\textit{Metric learning} is the task of learning an embedding (or feature vector) space, where similar objects are mapped closer to one another than dissimilar objects. In this setup, even objects unseen at training time can be categorised, by matching the learned representations against a support (reference) image set. In particular, in a \\textit{few-shot scenario}, the number of training examples and support images is kept to a minimum. Deep metric learning has been applied successfully to object recognition tasks \\cite{koch_siamese_2015,hoffer_deep_2015,schroff_facenet_2015}, even in real-world, robotic scenarios \\cite{zeng2018robotic}. Koch and colleagues \\cite{koch_siamese_2015} originally proposed to train two identical Convolutional Neural Networks (CNN) fed with images to be matched by similarity. This twin architecture is also known as Siamese Network. An extension of the Siamese architecture is the Triplet Network \\cite{hoffer_deep_2015,schroff_facenet_2015}, where the input data are fed as triplets including: (i) one image depicting a certain object class (i.e., \\textit{anchor}), (ii) a \\textit{positive example} of the same object, and (ii) a \\textit{negative example}, depicting a different object. The winning team for the object stowing task at the latest Amazon Robotic Challenge further tested the effects of learning weights independently on each CNN branch \\cite{zeng2018robotic}. Relaxing the weight-coupling constraint of Siamese and Triplet Networks was shown to benefit the matching of images across different visual domains, e.g., robot-collected images with product catalogue images. Hence, in what follows, we will use the two top-performing solutions in \\cite{zeng2018robotic} as a baseline to evaluate the object recognition performance of solutions which are purely based on Machine Learning. \n\n\\subsection{Hybrid Methods for Object Recognition}\nBroadly speaking, \\textit{hybrid reasoning methods} combine knowledge-based reasoning with Machine Learning. A detailed survey of hybrid methods developed to interpret the content of images can be found in \\cite{aditya_integrating_2019,gouidis_review_2020}. Many of these hybrid methods are specifically tailored on Deep NNs, which define the predominant approach to tackling object recognition problems. In this setup, background knowledge and knowledge-based reasoning can be integrated at four different levels of the NN \\cite{aditya_integrating_2019}: (i) in \\textbf{pre-processing}, to augment the training examples, (ii) within the \\textbf{intermediate layers}, (iii) as part of the \\textbf{architectural topology} or \\textbf{optimisation function}, and (iv) in the \\textbf{post-processing} stages, to validate the NN predictions.\n\nMethods in the first group rely on external knowledge to compensate for the lack of training examples. In \\cite{mancini_knowledge_2019}, auxiliary images depicting newly-encountered objects were first retrieved from the Web and then manually validated. As a result, significant supervision costs were introduced to compensate for the noisiness of data mined automatically.\n\nOther approaches have encoded the background knowledge directly in the inner layer representations of a Deep NN. In the RoboCSE framework, a set of knowledge properties of objects, (i.e., their typical location, fabrication material and affordances) were represented through multi-relational embeddings \\cite{daruna_robocse_2019}. This method was proven effective to infer the object's material, location and affordances from its class, but performed poorly on object categorisation tasks (i.e., when asked to infer the object's class from its properties).\n\nMore transparent and explainable than multi-relational embeddings, methods in the third group are either inspired by the topology of external knowledge graphs \\cite{marino_more_2017} or introduce reasoning components which are trainable end-to-end \\cite{serafini_logic_2016,manhaeve_deepproblog_2018,santoro_simple_2017,van_krieken_analyzing_2020}. Graph Search Neural Networks (GSNN) \\cite{marino_more_2017} resemble an input knowledge graph, where search seeds are selected based on a separate object detection module. In Logic Tensor Networks (LTN) \\cite{serafini_logic_2016}, entities are represented as distinct points in a vector space, based on a set of soft-logic assertions linking these entities. In this framework, symbolic rules (which may adhere to probabilistic logic - \\cite{manhaeve_deepproblog_2018}) are added as constraints to the NN's optimisation function. Authors in \\cite{santoro_simple_2017} proposed to incorporate reasoning in the form of a trainable Relational Reasoning Layer. Similarly, in \\cite{van_krieken_analyzing_2020}, differentiable knowledge statements (expressed in fuzzy logic) contribute to the training loss function, to aid digit classification on the MNIST dataset.\n\nFinally, the fourth family of hybrid methods uses knowledge-based reasoning to validate the object predictions generated through ML. In \\cite{young_towards_2016,young_semantic_2017} the results produced by the ML-based Vision module are first associated with the Qualitative Spatial Relationships extracted from the input image, and then also matched against the top-most related DBpedia concepts, if an unknown object is observed. As in the case of \\cite{santoro_simple_2017,van_krieken_analyzing_2020}, methods in this group can modularly interface with different NN architectures. Moreover, they make it possible to reason on objects unseen at training time, by querying external knowledge sources. For these reasons, in the approach proposed in this paper, knowledge-based reasoning is applied after generating the ML predictions. Because our focus is on reasoning about size, we cannot directly compare our approach against methods in \\cite{young_towards_2016,young_semantic_2017}, which focus on spatial reasoning and taxonomic reasoning (i.e., reasoning about the semantic relatedness of different object categories). \n\n\\subsection{Representing the Size of Objects}\nStudies on human visual cognition have suggested that there seems to be a set of preferred (or canonical) views which we use to mentally represent objects \\cite{rosch_principles_1999,hoffer_deep_2015}. These views have recognisable colour and shape features. Similarly, our perception of the object's sizes is influenced by a set of prototypical priors. Specifically, the \\textit{canonical size} we use to imagine and draw a certain object appears to be proportionally related to the logarithm of the object's assumed size, i.e., our \"prior knowledge about the size of objects in the world\" \\cite{konkle_canonical_2011}. Inspired by these findings, Bagherinezad et al. \\cite{bagherinezhad_are_2016} have modelled the object sizes through a log-normal distribution. The resulting distributions were then used to populate nodes in a graph, where objects which co-occurred frequently across the YFCC100M dataset \\cite{thomee_yfcc100m_2016} were linked together.\n\nThe size representation proposed in \\cite{bagherinezhad_are_2016} is both \\textit{quantitative} (i.e., expressed through statistical descriptors) and \\textit{qualitative }(i.e., smaller than or larger than, as symbolised by the graph's edges). Another quantitative representation was proposed in \\cite{elazar_how_2019}, where sizes (namely the object's length or volume) are represented as Distributions over Quantities (DoQ). In \\cite{zhu_reasoning_2014}, descriptors of the object's length are instead quantised with respect to three qualitative bins (i.e., <10in, 10\u2013100in and >100in). Compared to \\cite{zhu_reasoning_2014}, the statistical distributions in \\cite{bagherinezhad_are_2016,elazar_how_2019} can more expressively model the size variety within the same object class. Indeed, the same object class can comprise of many \\textit{instantiations} or \\textit{models} (a short novel and a dictionary are both books, although dictionaries are usually thicker). Moreover, the same object can be observed under different \\textit{appearances} (e.g., the book could be open or closed). Therefore, relying on a knowledge representation which can capture this within-class variability is crucial to ensure reuse across different real-world scenarios.\n\nAll the three approaches rely on data retrieved from the Web; this approach significantly reduces the cost of hardcoding a Knowledge Base about object sizes, but it is also more sensitive to noise. In addition, a limitation of the reviewed representations is that size is represented in one-dimensional terms, e.g., either through the object's volume or through its length. However, different physical dimensions (height, width and depth) contribute differently to characterising an object. For instance, recycling bins and coat stands may occupy a comparable volume, but bins are usually thicker than coat stands.\n\nThe size representations adopted in \\cite{bagherinezhad_are_2016,elazar_how_2019} have enabled to answer questions posed in natural language, such as \"are elephants bigger than butterflies?\". Furthermore, in \\cite{zhu_reasoning_2014}, the size features were used, among others, as an intermediate representation to predict the objects' \\textit{affordances}, or typical uses. Nonetheless, the role of size in object recognition is yet to be evaluated. In this paper, we propose a novel qualitative representation for the object's typical size, which requires minimal manual annotations and controls for the presence of noisy measurements.\n\n\\section{Methodology}\n\n\\subsection{Representing qualitative sizes in a Knowledge Base}\nWe identified 60 object categories which are commonly found in KMi, the setting in which we aim to deploy our robotic Health and Safety monitor, HanS. These include not only objects which are common to most office spaces (e.g., chairs, desktop computers, keyboards), but also Health and Safety equipment (e.g., fire extinguishers, emergency exit signs, fire assembly point signs), and objects which are, to some extent, specific to KMi (e.g., a foosball table, colorful hats from previous gigs of the KMi rock band, a KMi-branded welcome pod at the main entrance). The objective was then to associate each category in this catalogue to a series of typical size features, represented in qualitative terms. To this aim, we isolated three features contributing to the size of objects, namely their (i) \\textbf{front surface area} (i.e., the product of their width by their height), (ii) \\textbf{depth} dimension, and (iii) \\textbf{Aspect Ratio (AR)}, i.e., the ratio of their width to their height. With respect to the first dimension, we can characterise objects as \\textit{extra-small}, \\textit{small}, \\textit{medium}, \\textit{large} or \\textit{extra-large} respectively. Secondly, objects can be categorised as \\textit{flat}, \\textit{thin}, \\textit{thick}, or \\textit{bulky}, based on their depth. Thirdly, we can further discriminate objects based on whether they are \\textit{taller than wide} (\\textit{ttw}), \\textit{wider than tall} (\\textit{wtt}), or \\textit{equivalent} (\\textit{eq}), i.e., of AR close to 1. If the first two qualitative dimensions were plotted on a cartesian plane, a series of quadrants would emerge, as illustrated in Figure 1. Then, the AR can help to further separate the clusters of objects belonging to the same quadrant. For instance, doors and desks both belong to the extra-large and bulky group but doors, contrarily to desks, are usually taller than wide.\n\nHaving defined the cartesian plane of Figure 1, we can manually allocate the KMi objects to each quadrant and further sort the objects lying in the same quadrant. Sorting the objects manually ensures more reliable results than if the same information was retrieved automatically, especially given the paucity of resources encoding the relative size of objects \\cite{chiatti_towards_2020,bagherinezhad_are_2016}. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.2]{size_repr.pdf}\n  \\caption{Examples of object sorting across two dimensions: (i) the area of the front surface (the product of the width w by the height h), on the x axis; and (ii) the depth value d, on the y axis.}\n\\end{figure}\n\nMoreover, in the proposed representation, membership of each bin is mutually non-exclusive. Thus, with this representation, even classes which are extremely variable with respect to size, such as carton boxes and power cords, can be modelled. Indeed, boxes come in all sizes and power cords come in different lengths. Moreover, a box might lay completely flat, or appear bulkier, once assembled. Similarly, power cords, which are typically thinner than other pieces of IT equipment, might appear rolled up or tangled. \n\n\n\n\\subsection{Hybrid Reasoning Architecture}\n\nWe propose a modular approach to combining knowledge-based reasoning with Machine Learning for object recognition. In the proposed workflow, the knowledge of the qualitative size of objects (Section 3.1) is integrated in post-processing, after generating the ML-based object predictions. The proposed hybrid architecture is outlined in Figure 2 and consists of a series of sub-components, organised as follows. \n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\linewidth,trim=0 250 0 0, clip]{framework.pdf}\n  \\caption{The proposed architecture for hybrid object recognition. The knowledge-based reasoning module, which is aware of the typical size features of objects, can be modularly queried to validate the ML-based predictions.}\n\\end{figure}\n\n\\textbf{ML-based object recognition.} We rely on the state-of-the-art, ML-based object recognition methods of \\cite{zeng2018robotic}, to classify a set of pre-segmented object regions. Specifically, we classify objects by similarity to a reference image set, through a multi-branch Network. In this deep metric learning setting, predictions are ranked by increasing Euclidean (or L2) distance between each target embedding and the reference embedding set. Nevertheless, this configuration can be easily replaced by any other algorithm that provides, for each detected object, (i) a set of class predictions with an associated measure of confidence (whether similarity-based or probability-based) and (ii) the segmented image region enclosing the object. \n\n\\textbf{Prediction selection.} This checkpoint is conceived for assessing whether a ML-based prediction needs to be corrected or not. At the time of writing, we achieved good results simply by retaining those predictions which the ML algorithm is most confident about, and by running the remaining predictions through the size-based reasoner. Specifically, we avoid the knowledge-based reasoning steps if the top-1 class in the ML ranking: (i) has a ranking score smaller than $\\epsilon$ (i.e., the test image embedding lies near one of the reference embeddings, in terms of L2 distance); and also (ii) appears at least $i$ times in the top-K ranking. However, in Section 5, we also evaluate performance: (i) in the best case scenario where the ground truth labels are known and we know exactly which predictions to select for correction; and (ii) in the case where all ML-based predictions are passed to the size-based reasoner, without a pre-selection. \n\n\\textbf{Size estimation.} At this stage, the input depth image corresponding to each RGB scene is first converted to a 3D PointCloud representation. Then, statistical outliers are filtered out to reduce the impact of noisy points and extract the dense 3D region which best approximates the volume of the observed object. Specifically, all points which lie farther away than two standard deviations ($2\\sigma$) from their $n$ nearest neighbours are discarded. Because this outlier removal step is computationally expensive, especially for large 3D regions, we first sample each input PointCloud down so that 1 every $\\chi$ points is retained. Then, the Convex Hull algorithm is used to approximate the 3D box bounding the object region. Lastly, the x,y,z dimensions of the 3D bounding box are computed. Since the orientation of the object is not known a priori, we cannot unequivocally map any of the estimated dimensions to the object's real width and height. However, we can assume the object's depth to be the minimum of the returned x,y,z dimensions, due to the way depth measurements are collected through the sensor. Indeed, since we do not apply any 3D reconstruction mechanisms, we can expect that the measured depth underestimates the real depth occupied by the object.\n\n\\textbf{Size quantization.} The three dimensions obtained at the previous step are here expressed in qualitative terms. First, the two dimensions which were not marked as depth are multiplied together, giving a proxy of the object's surface area. The object is then categorised as extra-small, small, medium, large or extra-large, based on a set of cutoff thresholds $T$. Second, with respect to the estimated depth dimension, the object is categorised both as flat \/non-flat (based on a threshold $\\lambda_0$ ), and as flat, thin, thick, or bulky (based on a second set of thresholds $\\Lambda$, where $\\lambda_0 \\in \\Lambda$). Third, hypotheses are made about whether the object is taller than wide (ttw), wider than tall (wtt), or equivalent (eq), based on a cutoff $\\omega_0 \\in \\Omega$. It would be unfeasible to predict the object's Aspect Ratio from the estimated 3D dimensions, without knowing its current orientation. Therefore, we estimate the object's AR based on the width (w) and height (h) of the 2D box bounding the object region, as follows:\n\\begin{equation}\nAR = \\begin{cases}\nttw &\\text{if $h \\geq w \\land \\frac{h}{w} \\geq \\omega_0$}\\\\\nwwt &\\text{if $h<w \\land \\frac{w}{h} \\geq \\omega_0$}\\\\\neq &\\text{otherwise}\n\\end{cases}\n\\end{equation}\n\n\\textbf{Ranking validation.} The qualitative features returned by the size quantization module are then matched against the background KB introduced in Section 3.1, to identify the set of object categories which are plausible candidates for the observed object, based on their size. In Section 4, we test the effects of combining different size features (i.e., the area surface, thinness, and AR) to generate this candidate set. Ultimately, only those object classes in the original ML ranking which were validated as plausible with respect to the size are retained. \n\n\\section{Experiments}\n\\subsection{Data Preparation}\n\\subsubsection{KMi RGB Reference Set}\nFor training purposes, RGB images depicting 60 object classes commonly found in KMi were collected through a Turtlebot mounting an Orbbec Astra Pro RGB-Depth (RGB-D) monocular camera. Each object was captured against a neutral background and opportunely cropped to control for the presence of clutter and occluded parts. Coherently to our prior experimental setup \\cite{chiatti2020task}, we collected 10 images per class, with an 80\\%-20\\% training-validation split. Specifically, 5 images per class were used as anchor examples and paired up with their most similar image among the remaining 5 examples in that class (i.e., the multi-anchor switch strategy in  \\cite{zeng2018robotic}). Image similarity was assessed by: (i) extracting features from the pre-trained ResNet50 module, (ii) L2-normalising the resulting image embeddings, and (iii) computing their cosine similarity. Then, we also matched each anchor with the most similar image belonging to a different class. In this way, instead of picking negative examples randomly, i.e., the protocol followed in \\cite{zeng2018robotic}, we focused on triplets which are relatively harder to disambiguate.\n\n\\subsubsection{KMi RGB-D Test Set}\nFor testing purposes, additional RGB images and depth measurements of the KMi office environment were collected, during one of HanS' monitoring routines. As a result, the class cardinalities in this set reflect the natural, relative occurrence of objects in the observed domain. For instance, HanS is more likely to spot fire extinguishers and radiators than printers, on its scouting route. These data were initially recorded as Robot Operating System (ROS) bag files, with both RGB and depth messages logged at a 640x480 resolution (i.e., the maximum resolution allowed by the depth sensor). 622 clear RGB frames, where the robot camera was still, were manually selected from the original recording.\n\nBecause the recording of RGB and depth messages is asynchronous, each RGB frame had to be matched to its nearest depth image, within a time window of +\/- $\\mu$. Choosing a higher value for $\\mu$ increases the number of scenes for which a depth match is available, but also increases the chances that a certain scene is matched with the wrong depth measurements (e.g., because the robot has moved in the meantime). In our trials, we found that setting $\\mu$ to 0.2 seconds offered a good compromise. With this setup, a depth match was found for 509 (~82\\%) of the 622 original frames. After a visual inspection, only 2 (~0.4\\%) of the 509 depth matches were identified as inaccurate and discarded. This RGB-D set was further pruned to exclude identical images, where neither the robot's viewpoint nor the object arrangement had changed.\n\nWe annotated the remaining 213 images with respect to 60 reference object categories, through the Make Sense open-source annotation tool (https:\/\/www.makesense.ai\/). Besides labelling each object, we also annotated its bounding rectangular or polygonal region, depending on its 2D shape. Resorting to polygonal masks to segment non-rectangular or partially occluded objects allowed us to produce higher quality annotations and to mitigate the effects of marginal noise. These annotations were used to crop the original RGB images and store each object region as a separate test example. The same annotated regions were then reused to crop the depth image linked to each RGB frame. In this case, the output crop was stored as a 16bit grayscale PNG, where non-zero pixels encode the depth values in millimeters. Upon analysing the generated depth crop, we identified 19 object regions which did not enclose any depth measurement. This can happen, for instance, when the object falls outside the range of the depth sensor (i.e., 60 cm \u2013 8 m). To fairly compare the performance of the size-based reasoner (which relies on depth data) against the ML baseline, we discarded the empty depth crops from our test set, leaving us with a total 1414 object regions. \n\n\\subsection{Ablation Study}\nIn what follows, we list the different methods under evaluation and illustrate the changes introduced before each performance assessment.\n\n\\textbf{Baseline Nearest Neighbour (NN).} In this pipeline, feature vectors are extracted from a ResNet50 module pre-trained on ImageNet \\cite{he_deep_2016}, without re-training on the KMi RGB reference set. Namely, a 2048-dimensional embedding is extracted for each object region in the KMi RGB-D test set and matched to its nearest embedding in the KMi RGB reference set, in terms of L2 distance. This baseline provides us with a lower bound for the ML performance, before fine-tuning on the domain of interest.\n\n\\textbf{N-net} is the multi-branch Network which ensured the top performance on novel object classes, i.e., classes unseen at training time, in \\cite{zeng2018robotic}. In this configuration, each CNN branch is a ResNet50 module pre-trained on ImageNet. Image triplets are formed following the methodology discussed in Section 4.1.1. At training time, hyperparameters are updated so that the Triplet Loss is minimised. In this way, the Network optimisation is directed towards minimising the L2 distance between matching pairs, while also maximising the L2 distance between dissimilar pairs. At inference time, object regions in the KMi RGB-D test set are classified based on their nearest object in the KMi RGB reference set, as in the case of the baseline NN pipeline.\n\n\\textbf{K-net} is the multi-branch Network which led to the top performance on known object classes, i.e., classes seen at training time, in \\cite{zeng2018robotic}. K-net is a variation of N-net where a second loss component is added to the Triplet Loss. This auxiliary component of the loss derives from applying a SoftMax function to the last fully connected layer of the Network, to classify the input positive example over M classes. \n\n\\textbf{Hybrid (area).} This configuration follows the hybrid architecture introduced in Section 3.2. However, only the object's surface area is used as size feature to validate the ML predictions.\n\n\\textbf{Hybrid (area + flat).} With this ablation, we evaluate the effects of introducing a second feature to represent the size of objects. Specifically, here we consider not only the qualitative surface area of each object, but also whether they are flat or non-flat, based on their estimated depth. Then, the ML-based predictions are validated based on the set of object classes which both lie within the same area range and are also equally flat (or non-flat).\n\n\\textbf{Hybrid (area + thin)} is equivalent to the previous configuration, except the depth of objects is represented on a four-class scale: i.e., as flat, thin, thick, or bulky. The purpose of this ablation is testing the utility of introducing more granular bins in the y axis of Figure 1.\n\n\\textbf{Hybrid (area + flat + AR).} In this pipeline, we also integrate the qualitative Aspect Ratio (i.e., taller than wide, wider than tall, or equivalent) as a third knowledge property representing size. This setup is used for testing whether the qualitative AR can help further separating the object clusters exemplified in Figure 1.\n\n\\textbf{Hybrid (area + thin + AR) }follows the same configuration as the latter pipeline, except the object's depth is categorised as as flat, thin, thick, or bulky. \n\n\\subsection{Implementation Details}\n\n\\textbf{ML setup.} All the ML models illustrated in the previous Section were implemented in PyTorch \\cite{paszke_pytorch_2019}. Images in the KMi RGB reference set were resized to 224 \u00d7 224 frames and normalised to the same distribution as the ImageNet-1000 dataset, which was used for pre-training the ResNet50 modules in each CNN branch. The tested ablations were fine-tuned through an Adabound optimizer \\cite{luo_adaptive_2018}, over minibatches of 16 image triplets, with a learning rate set to start at 0.0001 and to trigger switching to Stochastic Gradient Descend optimization at 0.01. Parameters were updated for up to 1500 epochs, with an early stopping whenever the validation loss had not decreased for more than 100 epochs.\n\n\\noindent \\textbf{Knowledge-based reasoning parameters. }We relied on the Python Open3D library \\cite{zhou2018open3d} to process the PointCloud data and estimate the bounding 3D rectangles. During our trials we achieved the best results with threshold values set as follows. With respect to prediction selection, $\\epsilon$ was set to a distance of 0.04 and $i$ to 3 predictions, within a top-5 ranking, i.e., with $K=5$. The three cutoff sets in the size quantization module were defined so that $T=$\\{0.007,0.05,0.35,0.79\\} (with thresholds expressed in squared meters), $\\Lambda=$\\{0.1,0.2,0.4\\} (in meters) and $\\Omega$=\\{1.4 times\\}. \n\nAs an additional output of this work, we have also publicly released the RGB-D image set, annotated knowledge properties, and models used in these experiments: \\url{https:\/\/github.com\/kmi-robots\/object_reasoner}. \n\n\\section{Results and Discussion}\nPerformance on the KMi RGB-D test set was measured (i) in terms of the cross-class Accuracy, Precision (P), Recall (R) and F1 score of predictions in the top-1 result of the ranking; as well as (ii) based on the top-5 predictions in the ranking, in terms of mean Precision (P@5), mean normalised Discounted Cumulatve Gain (nDCG@5) and hit ratio (i.e., the ratio of the number of times the correct prediction appeared in the top-5 ranking to the total number of predictions). Specifically, the P, R and F1 metrics were aggregated across classes before and after weighing the averages by class support (i.e., based on the number of ground truth instances in each class). Measures@5 are unweighted. When comparing the different methods under evaluation, we prioritise improvements on the weighted F1 score, which accounts for the naturally imbalanced occurrence of classes in the test set. Moreover, at comparable top-1 results, we favour methods which provide higher quality top-5 rankings. Indeed, if the correct class was not ranked first but still appeared in the top-5 ranking, it would be easier for an additional reasoner (or human oracle) to correct the prediction.\n\n\\begin{table}\n  \\caption{ML baseline results on the KMi RGB-D test set.}\n  \\label{tab:ML}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Baseline NN & .33 &.36&.33&.25&.63&.33&.36&.23&.25&.54\\\\\n    N-net & .45 &.34&\\textbf{.40}&.31&.62&.45&.47&.33&.36&.63\\\\\n    K-net & \\textbf{.48} &\\textbf{.39}&\\textbf{.40}&\\textbf{.34}&\\textbf{.68}&\\textbf{.48}&\\textbf{.50}&\\textbf{.38}&\\textbf{.41}&\\textbf{.65}\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting only the wrong predictions.}\n  \\label{tab:best}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area) & .60&.52&.50&.47&.71&.60&.61&.43&.47&.72\\\\\n    Hybrid (area+flat) & .60&.55&.50&.48&.72&.60&.62&.44&.47&.72\\\\\n    Hybrid (area+thin) & .61&.55&.50&.48&.71&.61&.63&.44&.47&.71\\\\\n    Hybrid (area+flat+AR)& \\textbf{.62}&.59&.50&.49&\\textbf{.76}&\\textbf{.62}&\\textbf{.65}&\\textbf{.45}&\\textbf{.49}&\\textbf{.75}\\\\\n    Hybrid (area+thin+AR) & \\textbf{.62}&\\textbf{.62}&\\textbf{.51}&\\textbf{.52}&.74&\\textbf{.62}&\\textbf{.65}&\\textbf{.45}&.48&.74\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\nFirst, we evaluated all methods which are purely based on Machine Learning, i.e., before any background knowledge about the typical size of objects is integrated. The results of this first assessment are reported in Table 1. K-net is the ML baseline which led to the top performance, across all evaluation metrics. Therefore, we considered the K-net predictions as a baseline, for testing all the hybrid configurations.\n\nBecause the knowledge-based reasoner relies on different sub-modules (i.e., prediction selection, size estimation and ranking validation), and each one of these modules is likely to propagate its own errors, we initially tested performance assuming that the ground truth predictions are known and that we can accurately discern which ML predictions need to be corrected. Although unrealistic, this best-case scenario provides us with an upper bound for the reasoner's performance and aids the analysis of errors. As shown in Table 2, simply integrating knowledge about the qualitative surface area of objects already ensured a significant performance improvement, with a \\textbf{13\\%} increase of the unweighted F1 score and a \\textbf{11\\%} increase of the weighted F1 score. Overall, the best performance, both in terms of top-1 predictions as well as in terms of top-5 rankings, was achieved through the two hybrid configurations which included all the qualitative size features (i.e., surface area, thinness and AR). In particular, the unweighted F1 score increased up to \\textbf{18\\%} and the weighted F1 score up to \\textbf{15\\%}. Hence, the margin for improvement when complementing ML with size-based reasoning is significant. These results confirm the hypothesis laid in \\cite{chiatti_towards_2020}: the capability to compare objects by size and the access to background knowledge representing size play a crucial role in object categorisation. Notably, there is no significant difference between the results obtained when representing depth in binomial terms (i.e., as either flat or non-flat), as opposed to when more fine-grained categories are used (i.e., flat, thin, thick or bulky). Thus, we can hypothesise that the costs (and potential inaccuracies) associated with formalising additional priors for the objects' depth are not justified by a sufficient performance gain. \n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting all predictions.}\n  \\label{tab:worst}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area)& \\textbf{.49} &\\textbf{.34}&\\textbf{.33}&\\textbf{.29}&\\textbf{.59}&\\textbf{.49}&\\textbf{.50}&\\textbf{.37}&\\textbf{.40}&\\textbf{.62}\\\\\n    Hybrid (area+flat)& .47 &.33&.30&.27&.58&.47&.48&.37&.40&.60\\\\\n    Hybrid (area+thin)& .44 &.30&.25&.23&.53&.44&.44&.34&.37&.54\\\\\n    Hybrid (area+flat+AR)&.42 &.29&.22&.21&.58&.42&.44&.32&.35&.56 \\\\\n    Hybrid (area+thin+AR)& .39 &.28&.20&.19&.53&.39&.40&.30&.32&.50\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n  \\caption{Hybrid reasoning results, when correcting only an automatically selected subset of predictions.}\n  \\label{tab:final}\n  \\begin{tabular}{l*{10}{l}}\n    \\toprule\n    &  &\\multicolumn{3}{c}{Top-1 unweighted} & \\multicolumn{3}{c}{Top-1 weighted} & \\multicolumn{3}{c}{Top-5 results unweighted} \\\\\n    \\hline\n    Method & Top-1 Acc. & P & R & F1 & P & R & F1 & P@5 & nDCG@5 & Hit ratio \\\\ \n    \\midrule\n    Hybrid (area) & \\textbf{.55}&.42&\\textbf{.41}&\\textbf{.38}&.67&\\textbf{.55}&\\textbf{.57}&\\textbf{.43}&\\textbf{.46}&\\textbf{.69}\\\\\n    Hybrid (area+flat) & .54&\\textbf{.43}&.39&.37&.67&.54&.56&\\textbf{.43}&\\textbf{.46}&.68\\\\\n    Hybrid (area+thin) & .52&.39&.37&.35&.62&.52&.54&.42&.44&.64\\\\\n    Hybrid (area+flat+AR) & .53&\\textbf{.43}&.37&.36&\\textbf{.69}&.53&.56&\\textbf{.43}&.45&.68\\\\\n    Hybrid (area+thin+AR) & .51&\\textbf{.43}&.36&.36&.64&.51&.54&.41&.43&.63\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nThen, we ran all the predictions returned by the K-net method through the hybrid reasoning pipelines, without discriminating between correct and incorrect ML predictions (Table 3). This experiment is designed to evaluate the performance of the size-based reasoner in the worst-case scenario, i.e., when no assumption can be made about the ML algorithm confidence. As expected, the obtained results are comparable to or inferior to those achieved through a purely ML-based pipeline. These results indicate that applying the size priors indiscriminately led to inaccurate corrections of objects that the ML algorithm had correctly recognised. This issue is further emphasized the more size features (or knowledge priors) are introduced. As shown in Table 3, including other characteristics besides the object's surface area causes a performance drop.\n\nTherefore, to capitalise on the latent performance gains highlighted in Table 2, the ML and knowledge based outcomes need to be opportunely leveraged. To this aim, we introduced a meta-reasoning checkpoint (i.e., the prediction selection module described in Section 3.2) and automatically selected a subset of ML predictions to feed to the knowledge-based reasoner. The results of this last evaluation setup are summarised in Table 4. The tested hybrid configurations allowed to reach a compromise between the lower performance bound of Table 3 and the upper performance bound of Table 2. Specifically, the ML baseline was outperformed by up to \\textbf{4\\% }in terms of unweighted F1 and by up to \\textbf{7\\% } in terms of weighted F1. Moreover, introducing knowledge about the object's surface positively impacted the quality of the top-5 ranking: the mean P@5 and nDCG@5 both increased by \\textbf{5\\%}, and the hit ratio by \\textbf{4\\%}. Similarly to the results in Table 3, the qualitative surface area is the feature which led to the most consistent results across the different evaluation metrics. In other words, integrating additional knowledge beyond that first feature only led to comparable results, or even degraded the performance (i.e., in the case where a four-class scale instead of a binary one is used for the object's depth). Hence, in the experimental scenario of this paper, a size representation as minimalistic as indicating whether the object exposes an extra-small, small, medium, large, or extra-large front surface area is sufficient to ensure a significant boost in performance. \n\n\\section{Conclusion and Future Work}\nIn this paper we demonstrated that augmenting an object recognition pipeline based on Machine Learning with an effective representation of object sizes and a size reasoner can significantly improve the quality of predictions, as hypothesised in our prior work \\cite{chiatti_towards_2020}. These results are particularly promising, because they were achieved on image regions collected by a robot in its natural environment, i.e., in a more challenging experimental setup than the standard classification of benchmark image collections. In the proposed approach, we relied on a novel representation of the size of objects. Differently from prior knowledge representations, here we modelled size across three dimensions (the object's front surface area, depth and aspect ratio), to further separate the object clusters. Moreover, we allowed for annotating each object class with different size attributes, to adequately capture the size variability within each class.\n\nThe experiments presented in this paper also highlighted a series of directions of improvement, informing our future work. First, when estimating the object size from depth data we had to deal with hardware constraints: (i) objects falling outside the range of the depth sensors were excluded; (ii) highly reflective, absorptive or transparent materials (e.g., shiny metals, glass) altered the depth measurements. As such, access to a more advanced depth sensor would further improve the performance. Second, if the object was only partially visible in the original image, the estimated measurements (albeit accurate) would fail to represent the real object's size. Thus, incorporating the capability of moving towards the target object to refine the prediction through repeated measurements (i.e., Active Vision) is likely to benefit performance. Third, in the proposed method, we finetuned parameters through repeated trials. To increase the efficiency and scalability of our solution across different application scenarios, we will explore alternative solutions to (partially) automate this parameter tuning routine.\n\nNaturally, the relevance of background knowledge and knowledge-based reasoning for enabling Visual Intelligence spans way beyond the capability to reason about the typical size of objects. In \\cite{chiatti_towards_2020}, we have identified several other reasoners (e.g., spatial, compositional, motion-aware) which may enhance the robustness of state-of-the-art Machine Learning methods. Thus, in our future work, we will evaluate the performance impacts of integrating: (i) additional knowledge-based components, (ii) multiple sources of background knowledge, as well as (iii) effective meta-reasoning strategies, to reconcile the outcomes of different reasoners. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Statement of the problem and summary of results}\n\nLet $H$ be a matrix from the Gaussian unitary ensemble (GUE) and let each $G_i$ $(i=1,\\dots, M)$ denote a complex Ginibre matrix, i.e. a matrix with i.i.d. standard complex Gaussian entries. In this paper, we investigate the eigenvalues of the Hermitised product matrix\n\\begin{equation}\\label{W1}\nW_M = G_M^\\dagger \\cdots G_1^\\dagger H G_1 \\cdots G_M\n\\end{equation}\nunder the assumption that all matrices, $H$ and $G_i$ ($i=1,\\ldots,M$), are independent. We will see that the eigenvalues form a bi-orthogonal ensemble~\\cite{Bo98}. Furthermore, this ensemble is closely related (in a sense that will be specified in the next subsection) to the so-called Hermite Muttalib--Borodin ensemble~\\cite{Mu95,Bo98}. The latter is defined by the joint eigenvalue probability density function (PDF)\n\\begin{equation}\\label{MB-Hermite}\n\\tilde P(x_1,\\ldots,x_N)=\\frac1{\\tilde Z_N^{(M)}}\n\\prod_{1\\leq j<k\\leq N}(x_k-x_j)(x_k^{2M+1}-x_j^{2M+1})\n\\prod_{\\ell=1}^N|x_\\ell|^\\alpha e^{-x_\\ell^2},\n\\end{equation}\nwhere $\\tilde Z_N^{(M)}$ is a normalisation constant and $\\alpha$ is a non-negative constant.\n\nIt will transpire that the bi-orthogonal ensemble structure associated with the eigenvalue PDF of the product matrix~\\eqref{W1} is a corollary of the following more basic result.\n\n\\begin{thm}\\label{T1}\nLet $G$ be an $n \\times N$ ($n \\le N$) standard complex Gaussian matrix and let $A$ be an $n \\times n$\nHermitian matrix with eigenvalues $a_1,\\ldots,a_N$.\nIf the eigenvalues of $A$ are pairwise distinct and ordered as\n\\begin{equation}\\label{as}\na_1<a_2<\\cdots<a_{n_0}<0<a_{n_0+1}<\\cdots<a_n,\n\\end{equation}\nthen the PDF of the non-zero eigenvalues of matrix $X=G^\\dagger AG$ is given by\n\\begin{multline}\\label{as1}\nP^{n_0}_{n}(\\{a_j\\}_{j=1}^n;\\{x_j\\}_{j=1}^n)=\\\\\n\\prod_{l=1}^n\\frac1{|a_l|}\\frac{(x_l\/a_l)^{N-n}}{(N-l)!}\n\\prod_{1\\leq j<k\\leq n}\\frac{x_k-x_j}{a_k-a_j}\\det\\big[e^{-x_i\/a_j}\\big]_{i,j=1}^{n_0}\n\\det\\big[e^{-x_{i}\/a_{j}}\\big]_{i,j=n_0+1}^{n},\n\\end{multline}\nwhere\n\\begin{equation}\\label{as2}\nx_1< \\cdots < x_{n_0} < 0 < x_{n_0+1} < \\cdots < x_n.\n\\end{equation}\nIn particular, we see that $X$ has $n_0$ ($n-n_0$) negative (positive) eigenvalues, i.e. the same number as $A$.\nThe remaining $N-n$ eigenvalues are all identically zero.\n\\end{thm}\n\n\nWe remark that in Theorem \\ref{T1} the case of $n<N$ remains unanswered although  this  is certainly  of high  interest; see \\cite{DF06,ACLS} and references therein for a relevant  question.   \n\n\n\n\nThe rest of this paper is organised as follows:\nIn Section~\\ref{sec:product}, we use Theorem~\\ref{T1} to find the PDF for the eigenvalues of the product~\\eqref{W1} as a bi-orthogonal ensemble. Moreover, the explicit expression for the PDF is seen to reduce asymptotically to the functional form~\\eqref{MB-Hermite} specifying the Hermite Muttalib--Borodin ensemble.\nExplicit expressions for the bi-orthogonal functions are derived in Section~\\ref{sec:biortho}. Analogous to the theory of Hermite polynomials (see e.g.~(\\ref{HL}) below), we will see that it is convenient to consider bi-orthogonal functions of even and odd degree separately.\nSection~\\ref{sec:int} provides reformulations of the bi-orthogonal functions and the correlation kernel as integral representations, which are more suited for asymptotic analysis. These integral representations can also be expressed in terms of Meijer $G$-functions, and we will see that they are closely related to known formulae stemming from the product ensemble of Laguerre type.\nThe local scaling limit at the origin is derived and seen to be related to the Meijer $G$-kernel in Section~\\ref{sec:hard}. This result is also compared with the local scaling limit of the Hermite Muttalib--Borodin ensemble.\nSection~\\ref{sec:global} includes derivations of the global spectrum of the product~\\eqref{W1} as well as the Hermite Muttalib--Borodin ensembles. Since our product ensemble reduces asymptotically to the Hermite Muttalib--Borodin ensemble, they are as expected seen to have the same global spectrum, which in turn is given in terms of the Fuss--Catalan density.\nFinally, Theorem~\\ref{T1} is proven in the appendix. This theorem is an important result by itself. For this reason, we provide three separate proofs each with their own merits.\n\n\n\n\n\\subsection{First motivation: Muttalib--Borodin ensembles}\n\\label{sec:motivation}\n\nOrthogonal polynomial ensembles are point processes on (a subset of) the real line with a joint distribution given by\n\\begin{equation}\\label{jpdf-classical}\nP(dx_1,\\ldots,dx_n)=\\frac1{Z_n}\\Delta_n(\\{x\\})^2\\prod_{k=1}^nw(x_k)dx_k,\n\\end{equation}\nwhere $Z_N$ is a normalisation constant, $w(x)$ is a non-negative weight function, and $\\Delta_n(\\{x\\})$ denotes the Vandermonde determinant,\n\\begin{equation}\n\\Delta_{n}(\\{x\\})=\\det_{1\\leq i,j\\leq n}\\big[x_i^{\\,j-1}\\big]=\\prod_{1\\leq i<j\\leq n}(x_j-x_i).\n\\end{equation}\nLike the corresponding moment problem, it is often useful to distinguish between models with support on a finite, semi-infinite, and double-infinite interval. The canonical examples are the Jacobi-, Laguerre-, and Hermite-ensembles summarised in Table~\\ref{table:weights}. These ensembles are named according to the corresponding classical orthogonal polynomials. In fact, for $a\\neq0$ the latter would more appropriately be called the generalised Hermite ensemble.\n\\begin{table}[htbp]\n\\centering\n\\caption{Summary of weight functions and support for the three canonical orthogonal ensembles in random matrix theory given by the joint distribution~\\eqref{jpdf-classical}.}\n\\vspace*{.5em}\n\\label{table:weights}\n\\begin{tabular}{l@{\\qquad\\qquad}l@{\\qquad\\qquad}l}\n\\hline\\hline\nensemble  & \\hspace*{1.5em}weight & support\\\\ \\hline \\\\[-.9em]\nJacobi & $w(\\lambda)=\\lambda^a(1-\\lambda)^b$  & $\\lambda\\in(0,1)$ \\\\[.2em]\nLaguerre & $w(\\lambda)=\\lambda^ae^{-\\lambda}$ & $\\lambda\\in(0,\\infty)$ \\\\[.2em]\nHermite & $w(\\lambda)=|\\lambda|^ae^{-\\lambda^2}$ & $\\lambda\\in(-\\infty,\\infty)$\\\\[.1em] \\hline\\hline\n\\end{tabular}\n\\end{table}\n\nIn random matrix theory these three ensembles play a fundamental role as they appear as the distribution of the eigenvalues (or singular values) for the transfer (or truncated unitary) ensemble, the complex Wishart (or chiral) ensemble, and the Gaussian unitary ensemble, respectively; see e.g.~\\cite{Fo10}.\n\nA fundamental insight, which can be traced back to Wigner~\\cite{Wi57}, is that the joint distribution~\\eqref{jpdf-classical} allows an interpretation as the equilibrium measure for a one-dimensional gas of pairwise repulsive point particles in a confining potential.\nMore precisely, consider the Gibbs measure for a classical gas of $n$ point particles which are pairwise repulsive according to a two-point potential $U(x,y)$ and confined by a common one-point potential $V(x)$, i.e.\n\\begin{equation}\\label{gibbs}\nP(dx_1,\\ldots,dx_n)=\\frac{1}{Z_n}e^{-\\beta E(x_1,\\ldots,x_n)}\\prod_{k=1}^n dx_k\n\\end{equation}\nwith $\\beta$ denoting the inverse temperature and $E$ the energy functional\n\\begin{equation} \\label{H1}\nE(x_1,\\ldots,x_n)=\\frac12\\sum_{k=1}^n V(x_k)-\\sum_{1\\leq i<j\\leq n}U(x_j,x_i).\n\\end{equation}\nWe see that at $\\beta=2$ (sometimes referred to as the free fermion point) the Gibbs measure~\\eqref{gibbs} is identical to~\\eqref{jpdf-classical} provided we set $V(\\lambda)=-\\log w(\\lambda)$ and $U(x_i,x_j)=\\log|x_j-x_i|$. In this way, the eigenvalues of random matrices relate to the Boltzmann factor of a simple statistical mechanical system with one- and two-body interactions only.\n\nA recent development in random matrix theory is the study of exactly solvable product ensembles; see~\\cite{AI15} for a review.\nAs an example, let $G_i$ ($i=1,\\ldots,M$) be independent complex Ginibre matrices (matrices whose entries are i.i.d. standard complex Gaussians)\nand consider the Hermitian product\n\\begin{equation}\\label{old-product}\nG_{M}^\\dagger\\cdots G_1^\\dagger G_1\\cdots G_M.\n\\end{equation}\nFrom the work of Akemann et al.~\\cite{AKW13,AIK13}, we know that the explicit\nPDF for the eigenvalues of the matrix~\\eqref{old-product} is\n\\begin{equation}\\label{2}\nP^{(M)}(x_1,\\ldots,x_n)=\\frac1{Z_n^{(M)}}\\Delta_n(\\{x\\})\\det_{1\\leq i,j\\leq n}\\big[g_j^{(M)}(x_i)\\big],\n\\qquad x_i > 0 \\: (i=1,\\dots,n)\n\\end{equation}\nwhere $Z_n^{(M)}$ is a known normalisation constant and $g_j^{(M)}(x)$ ($j=1,\\ldots,n$) are given by certain Meijer $G$-functions.\nGenerally, such PDFs are known as polynomial ensembles~\\cite{KS14}.\n\nIt seems natural to ask whether these product ensembles also have (at least approximately) an interpretation as a Gibbs measure of the form~\\eqref{gibbs} and~\\eqref{H1}.\nHowever, unlike the Vandermonde determinant, the determinant in~\\eqref{2} cannot be evaluated as a product (for $M \\ge 2$).\nThis prohibits a literal interpretation of the eigenvalues of~\\eqref{old-product} as a statistical mechanical system\nwith only one- and two-body interactions. One could fear that this meant that there was no simple physical interpretation related to~\\eqref{2}.\nHowever, if we consider (\\ref{2}) with each $x_j$ large, the Meijer $G$-functions can be replaced by their asymptotic approximation~\\cite{Fi72}.\nAfter a change of variables, the joint density~\\eqref{2} to leading order in the asymptotic expansion becomes~\\cite{FLZ15}\n\\begin{equation}\\label{MB-laguerre}\n\\tilde P^{(M)}(x_1,\\ldots,x_n)=\n\\frac1{\\tilde Z_n^{(M)}}\\Delta_n(\\{x\\})\\Delta_n(\\{x^M\\})\\prod_{k=1}^n x_k^a\\,e^{-x_k},\n\\qquad x_k > 0 \\: (k=1,\\dots,n)\n\\end{equation}\nwhere $a$ is a known non-negative constant.\nThis does correspond to the Boltzmann factor of a statistical mechanical system with one- and two-body interactions only.\n\nA comparison between~\\eqref{2} and~\\eqref{MB-laguerre} can be done a posteriori.\nA connection between the two ensembles was first noted by Kuijlaars and Stivigny~\\cite{KS14}, who observed that the hard edge scaling limit of~\\eqref{MB-laguerre} found in~\\cite{Bo98} took the same functional form as the Meijer $G$-kernel found in the product ensemble~\\cite{KZ14}, albeit with a different choice of parameters. Due to recent progress, even more is known about the scaling limits of both models, and their similarities. Thus it has been established that the two ensembles also share the same global spectral distribution~\\cite{Mu02,BJLNS10,BBCC11,PZ11,FW15}. Furthermore, in both cases the local correlations in the bulk and near the soft edge are given by the familiar sine and Airy process, respectively~\\cite{LWZ14,Zh15}.\n\nThe ensemble~\\eqref{MB-laguerre} had, in fact, appeared in earlier random matrix literature.\nIt was first isolated by Muttalib~\\cite{Mu95}, who suggested it as a naive approximation to the transmission eigenvalues in a problem about quantum transport.\nA feature of the new interaction is that bi-orthogonal polynomials (rather than orthogonal polynomials) are needed in the study of correlation functions. Such bi-orthogonal ensembles were considered in greater generality by Borodin~\\cite{Bo98}, who devoted special attention to PDFs\n\\begin{equation}\\label{MB}\nP(x_1,\\ldots,x_n)=\\frac1{Z_n}\\prod_{j=1}^n w(x_l)\\prod_{1\\leq j<k\\leq n} \\big|x_k-x_j\\big|\\,\n\\big|\\sgn(x_k)|x_k|^{\\theta}-\\sgn(x_j)|x_j|^{\\theta}\\big|,\n\\end{equation}\nwith $\\theta>0$ and $w(x)$ representing one of the three classical weight functions from Table~\\ref{table:weights}.\nFollowing \\cite{FW15}, we will henceforth refer to these ensembles as the (Jacobi, Laguerre, Hermite) Muttalib--Borodin ensembles.\nWe note that the awkward dependence of signs in the last factor in~\\eqref{MB} disappears when the eigenvalues are non-negative (e.g. for Laguerre- and Jacobi-ensembles) and when $\\theta$ is an odd integer as in~\\eqref{MB-Hermite}.\n\nAt the time of their introduction,\nthe Muttalib--Borodin ensembles had no obvious relation to any random matrix models defined in terms of PDFs on their entries (except for the trivial case $\\theta=1$), and could merely be interpreted as a simple one-parameter generalisation of the classical ensembles.\nHowever, we now see that the Laguerre Muttalib--Borodin ensemble has a close connection\nto products of complex Gaussian random matrices~\\eqref{old-product} through the approximation~\\eqref{MB-laguerre}.\n\nKnowing that the Laguerre Muttalib--Borodin ensemble appears as an asymptotic approximation to the Gaussian product~\\eqref{old-product}, it seems natural to ask the reverse question: \\emph{Can we find product ensembles which reduce asymptotically to the Jacobi and Hermite Muttalib--Borodin ensembles?} If this is possible, it would be reasonable to say we have completed a link between the Muttalib--Borodin ensembles with classical weights and the new family of product ensembles.\n\nFor the Jacobi Muttalib--Borodin ensemble a link to products of random matrices is provided by looking at the squared singular values of a product of truncated unitary matrices~\\cite{KKS15,FW15}. In this paper, it is our aim to isolate a random matrix product structure for which the eigenvalue PDF reduces asymptotically to the functional form of the Hermite Muttalib--Borodin ensemble. This construction therefore completes the correspondence between product ensembles and the three Muttalib--Borodin ensembles with classical weights, i.e. Laguerre, Jacobi, Hermite. Furthermore, the relevant product ensemble provides by itself a new interesting class of integrable models, which unlike all previous product ensembles (see~the review \\cite{AI15}) allows for negative eigenvalues.\n\nAs the product ensemble in question must allow for negative eigenvalues, it is no longer sufficient to investigate Wishart-type matrices like~\\eqref{old-product} which are positive-definite by construction.\nIt turns out that the correct structure is the Hermitised product of a GUE matrix and $M$ complex Ginibre matrices given by~\\eqref{W1}.\nThe case $M = 1$ of~\\eqref{W1} has previously been isolated in the recent paper of Kumar~\\cite{Ku15} as an example\nof a matrix ensemble which permits an explicit eigenvalue PDF.\n\n\\subsection{Second motivation: hyperbolic Harish-Chandra--Itzykson--Zuber integrals}\n\nAnother reason that the Hermitised random matrix product~\\eqref{W1} is of particular interest is its relation to the so-called hyperbolic Harish-Chandra--Itzykson--Zuber (HCIZ) integral. By way of introduction on this point, we note that it is by now evident that the family of exactly solvable product ensembles is intimately linked to a family exactly solvable group integrals sometimes referred to as integrals of HCIZ type. For the study of products of Ginibre matrices~\\eqref{old-product} it was sufficient to know the familiar (and celebrated) HCIZ integral~\\cite{HC57,IZ80}:\n\\begin{equation}\\label{HCIZ}\n \\int_{U(N)\/U(1)^N}e^{-\\tr AVBV^{-1}}\\,(V^{-1}dV)=\\pi^{N(N-1)\/2}\n \\frac{\\det[e^{-a_ib_j}]_{i,j=1}^{N}}\n {\\prod_{1\\leq i<j\\leq N} (a_j-a_i)(b_j-b_i)},\n\\end{equation}\nwhere $(V^{-1}dV)$ denotes the Haar measure on the unitary quotient group $U(N)\/U(1)^N$, while $A$ and $B$ are Hermitian $N\\times N$ matrices with eigenvalues $a_1<\\cdots<a_N$ and $b_1<\\cdots<b_N$, respectively. However, for studies of products of spherical, truncated unitary, or coupled random matrices generalisations of the HCIZ integral are needed, see~\\cite{KKS15,AS16,Liu17} for the two latter cases. We emphasise that the product of truncated unitary matrices considered by Kieburg et al.~\\cite{KKS15} required a previously unknown generalisation of the HCIZ integral. Likewise, our study of the Hermitised random matrix product~\\eqref{W1} requires knowledge about the so-called hyperbolic HCIZ integral in which the integration on the left-hand side of~\\eqref{HCIZ} should be replaced with an integration over the pseudo-unitary group (see Section~\\ref{sec:fyodorov} for details). The study of such hyperbolic group integrals was initiated by Fyodorov~\\cite{Fy02,FS02}.\nAn interesting feature of the hyperbolic HCIZ integral is that the integration over the pseudo-unitary is non-compact, which forces us to introduce some additional constraints on the Hermitian matrices $A$ and $B$ to ensure convergence; this is a difficulty which does not arise in other HCIZ-type integrals. Finally, we mention that HCIZ-type integrals have other applications in theoretical and mathematical physics beyond products of random matrices, e.g.~the hyperbolic HCIZ integral was used to find the spectral properties of the Wilson--Dirac operator in lattice quantum chromodynamics~\\cite{KVZ13}. Moreover, HCIZ-type integrals represent a rich area of mathematical research, for example within the study of Lie groups, harmonic analysis, combinatorics, and probability (e.g. matrix-valued Brownian motion); see\ne.g.~the text \\cite{Te88a}.\n\n\n\\section{Products of random matrices and Hermite Muttalib--Borodin ensembles}\n\\label{sec:product}\n\nIn this section, we establish that the eigenvalue PDF of the matrix product~\\eqref{W1}\nis a polynomial ensemble and show that it reduces asymptotically to the Hermite Muttalib--Borodin ensemble~\\eqref{MB-Hermite}.\n\nAs stated in the introduction, the eigenvalue PDF of~\\eqref{W1} follows as a consequence of Theorem~\\ref{T1}. The idea is simple: let $A$ be a random matrix from a polynomial ensemble, i.e. it has an eigenvalue PDF of the form\n\\begin{equation}\\label{af}\nP_A(\\{a_k\\}_{k=1}^n)=\\frac1{Z_n}\\prod_{1\\leq i<j\\leq n}(a_j-a_i)\\det[w_j(a_i)]_{i,j=1}^n,\n\\end{equation}\nwhere $a_1\\leq a_{2}\\leq \\cdots\\leq  a_{n}$ are the (ordered) eigenvalues of $A$, $w_j$ ($j=1,\\ldots,n$) is a family of weight functions, and $Z_n$ is a normalisation constant. Now, let $G$ be an $n \\times N$ $(n \\le N)$ standard complex Gaussian matrix. Then Theorem~\\ref{T1} gives the eigenvalue PDF of $G^\\dagger AG$. Moreover, it is seen that this new eigenvalue PDF is also a polynomial ensemble. In other words, Theorem~\\ref{T1} provides a map from the class of polynomial ensembles into itself. Thus, we may apply Theorem~\\ref{T1} recursively to construct hierarchies of polynomial ensembles.\nLet us make this statement more precise.\n\n\\begin{lemma}\\label{C1}\nLet $G$ be an $n \\times N$ $(n \\le N)$ standard complex Gaussian matrix, and let $A$ be a random matrix from a\npolynomial ensemble with eigenvalue PDF (\\ref{af}), independent of $G$.\nThen the PDF for the non-zero eigenvalues of the random matrix product $G^\\dagger AG$ is equal to\n\\begin{equation}\\label{af2}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det_{1\\leq i,j\\leq n}\\bigg[\\int_0^\\infty \\frac{da\\,e^{-a}}{a^{n-N+1}}\\,w_{j}\\big(\\frac{x_i}{a}\\big)\\bigg]\n\\end{equation}\nwith the eigenvalues ordered  $x_1\\leq x_{2}\\leq \\cdots\\leq  x_{n}$.\n\\end{lemma}\n\n\n\\begin{proof}\nIn order to use Theorem~\\ref{T1}, we fix an $n_0\\in\\{0,1,\\ldots,n\\}$ and assume that the eigenvalues of $A$ are ordered as~\\eqref{as}.\nConsequently, the non-zero eigenvalues of $G^\\dagger AG$  can be ordered as~\\eqref{as2} almost surely.\nIt follows from the conditional eigenvalue PDF~\\eqref{as1} and~\\eqref{af} that the eigenvalue PDF of $G^\\dagger AG$ (up to $N-n$ eigenvalues which are identically zero) is given by\n\\begin{multline}\\label{recurrence-step}\n\\int_D P_A(\\{a_k\\}_{k=1}^n)P^{n_0}_{n}(\\{a_j\\}_{j=1}^n;\\{x_j\\}_{j=1}^n)\\, da_1\\cdots da_n\n=\\frac1{Z_n}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\\\\\n\\times\\int_D\\prod_{l=1}^n\\frac1{|a_l|}\\frac{(x_l\/a_l)^{N-n}}{(N-l)!}\n\\det\n\\begin{bmatrix}\n \\{e^{-x_i\/a_j}\\}_{i,j=1}^{n_0} & 0 \\\\\n 0 & \\{e^{-x_{i}\/a_{j}}\\}_{i,j=n_0+1}^{n}\n\\end{bmatrix}\n\\det[w_j(a_i)]_{i,j=1}^n\n\\, da_1\\cdots da_n,\n\\end{multline}\nwhere the domain of integration $D$ is given according to~\\eqref{as} and the eigenvalues $x_i$ ($i=1,\\ldots,n$) are ordered according to~\\eqref{as2}.\nWe note that the integral on the second line in~\\eqref{recurrence-step} is a close cousin to the well-known Andreief integral~\\cite{An83,Br55}.\nUpon expansion of the determinants, it is readily seen that~\\eqref{recurrence-step} may rewritten as\n\\begin{equation}\\label{recurrence-step2}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det\n\\begin{bmatrix}\n \\displaystyle\\bigg\\{-\\int_{-\\infty}^0\\frac{da}a (x_i\/a)^{N-n} e^{-x_i\/a}\n w_j(a)\\bigg\\}_{\\substack{i=1,\\ldots,n_0\\\\j=1,\\ldots,n}} \\\\\n \\displaystyle\\bigg\\{\\int^{\\infty}_0\\frac{da}a (x_i\/a)^{N-n} e^{-x_i\/a}\n w_j(a)\\bigg\\}_{\\substack{i=n_0+1,\\ldots,n\\\\j=1,\\ldots,n}}\n\\end{bmatrix}.\n\\end{equation}\nIf we make a change of variables $a\\mapsto -a$ in the first $n_0$ rows in the determinant in~\\eqref{recurrence-step2} then we get\n\\begin{equation}\n\\frac1{Z_n}\\prod_{l=1}^n\\frac{1}{(N-l)!}\\prod_{1\\leq j<k\\leq n}(x_k-x_j)\n\\det_{1\\leq i,j\\leq n}\\bigg[\\int_0^\\infty \\frac{da}a\\,\\frac{e^{-|x_i|\/a}}{(|{x_i}|\/a)^{n-N}}\\,w_j((\\sgn x_i)a)\\bigg];\n\\end{equation}\nrecall that $x_i$ ($i=1,\\ldots,n$) is ordered according~\\eqref{as2}. Finally, if we make another change of variables $a\\mapsto |x_i|\/a$ in the $i$-th row, then~\\eqref{af2} follows for any fixed $n_0$. Note that this result is independent of the choice of $n_0$, so we have the final result.\n\\end{proof}\n\n\\begin{remark}\nThe study of maps from the space of polynomial ensembles onto itself is an interesting endeavour, since such maps give rise to new random matrix ensembles without destroying integrability. In fact, the study of such maps is presently an active area of research in random matrix theory~\\cite{CKW15,Ku16,KK16,KR16}. Lemma~\\ref{C1} provides a new transformation to this class of maps, which cannot be obtained directly from any of the previously established transformations. We note that Lemma~\\ref{C1} includes the transformation~\\cite[Theorem 2.1]{KS14} as a special case arising when the matrix $A$ is positive definite. A restriction of Lemma~\\ref{C1} is that $n\\leq N$. Thus it is seen that the PDF~\\eqref{af2} develops a singularity for $n>N$ indicating that the formula is no longer generally valid in this case, depending on the properties of $w_j$. It would be interesting to extend the above results to include the case $n>N$ more generally.\n\\end{remark}\n\nWith Lemma~\\ref{C1} at hand, we are ready to write down the eigenvalue PDF for the product~\\eqref{W1}.\n\n\\begin{thm}\\label{cor-matrix}  Let $\\nu_0=0, \\nu_1,  \\ldots, \\nu_M$ be non-negative integers.\nSuppose that\n$H$ is an $n\\times n$ GUE matrix and  $G_1, \\ldots, G_M$  are independent  standard complex Gaussian matrices where  $G_m$ is of size $(\\nu_{m-1}+n) \\times (\\nu_{m} +n)$.\nThen the joint PDF for the non-zero eigenvalues of the matrix~\\eqref{W1} is given by\n\\begin{equation}\\label{PDF-matrix}\nP^{(M)}(x_1,\\ldots,x_n)=\\frac{1}{Z^{(M)}_n}\\prod_{1\\leq i<j\\leq n}(x_j-x_i)\\det_{1\\leq i,j\\leq n}\\big[g_{j-1}^{(M)}(x_i)\\big],\n\\end{equation}\nwhere the weight functions $g_j^{(M)}$ are defined recursively by\n\\begin{equation}\\label{weight-recursive}\ng_j^{(0)}(x)=x^{j}e^{-x^2}\n\\quad\\text{and}\\quad\ng_j^{(m)}(x)=\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_m}e^{-y}\\,g_j^{(m-1)}(x\/y),\\quad m=1,\\ldots,M\n\\end{equation}\nand the normalisation constant is\n\\begin{equation}\nZ^{(M)}_n={2^{-n(n-1)\/2}\\pi^{n\/2}}\\prod_{m=0}^M\\prod_{j=1}^n\\Gamma(\\nu_m+j).\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nFirst, let us consider the simplest situation, that is a product of square matrices, i.e. $\\nu_1=\\cdots=\\nu_M=0$. As the eigenvalue PDF of an $n\\times n$ GUE matrix is given by~\\eqref{PDF-matrix} with $M=0$, the theorem follows immediately by applying Lemma~\\ref{C1} $M$ times.\n\nWe need to be a little more careful when the case of rectangular matrices is considered.\nThe $M=1$ case of the theorem is again an immediate consequence of Lemma~\\ref{C1}, which gives us the eigenvalues of $W_1=G_1^\\dagger HG_1$. However, in order to apply Lemma~\\ref{C1} a second time and find the non-zero eigenvalues of $W_2=G_2^\\dagger W_1G_2$, we have to take into account that $W_1$ has a zero eigenvalue with multiplicity $\\nu_1$. To proceed, we can use the same idea as in~\\cite{IK14}. The unitary invariance of Gaussian matrices tells us that $W_1\\stackrel{d}=U^\\dagger W_1 U$ and $G_2\\stackrel{d}{=}VG_2$ for any $U,V\\in U(n+\\nu_1)$. It thus follows\n\\begin{equation}\\label{reduction}\nW_2=G_2^\\dagger W_1 G_2\\stackrel{d}{=}\n\\begin{bmatrix} \\tilde G_2^\\dagger & g_2^\\dagger \\end{bmatrix}\n\\begin{bmatrix} X_1 & 0 \\\\ 0 & 0 \\end{bmatrix}\n\\begin{bmatrix} \\tilde G_2 \\\\ g_2 \\end{bmatrix}\n=\\tilde G_2^\\dagger X_1 \\tilde G_2,\n\\end{equation}\nwhere $X_1=\\diag(x_1,\\ldots,x_n)$ is an $n\\times n$ diagonal matrix distributed according to~\\eqref{PDF-matrix} with $M=1$, while $\\tilde G_2$ and $g_2$ are standard Gaussian matrices of size $n\\times (n+\\nu_2)$ and $\\nu_1\\times (n+\\nu_2)$, respectively. Now, Lemma~\\ref{C1} can be applied to the right-hand side in~\\eqref{reduction}, which gives us the PDF of the non-zero eigenvalues of $W_2$. Repeating this procedure completes the proof.\n\\end{proof}\n\n\\begin{remark}\nWe note that the case $M=1$ of Theorem~\\ref{cor-matrix} is in agreement with the result stated by Kumar~\\cite[Eq.~(46) and~(47)]{Ku15}. However, the derivation therein is incomplete due to the reliance on the HCIZ integral\n(1.14), rather than its hyperbolic variant (A.38) below.\n\\end{remark}\n\nThere are many other representations for the weight functions in Theorem~\\ref{cor-matrix}\nbeyond the recursive definition~\\eqref{weight-recursive}.\nAs usual, it is particularly useful for analytic purposes to write the weight functions in their contour integral representation.\n\n\\begin{lemma}\nWe have\n\\begin{equation}\\label{contour-matrix}\ng_j^{(M)}(x)=\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^M\\Gamma(\\nu_m-s),\n\\end{equation}\nwhere $c<0$ is a negative constant.\n\\end{lemma}\n\n\\begin{proof}\nBy means of the residue theorem, it is seen that\n\\begin{equation}\ng_j^{(0)}(x)=x^{j}e^{-x^2}=\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big).\n\\end{equation}\nNow, assume that $g_j^{(M-1)}(x)$ is given by~\\eqref{contour-matrix}. From the recursive formula~\\eqref{weight-recursive}, we have\n\\begin{equation}\ng_j^{(M)}(x)=\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_M}e^{-y}\\,\\frac{(\\sgn x)^{j}}{4\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Big(\\frac{|x|}y\\Big)^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^{M-1}\\Gamma(\\nu_m-s).\n\\end{equation}\nIt is a straightforward exercise, considering the asymptotic decay, to show that\nwith $c<0$ the order of the integrals may be interchanged.   Thus we have\n\\begin{equation}\ng_j^{(M)}(x)=\n\\frac{(\\sgn x)^{j}}{4\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,|x|^s\\,\\Gamma\\Big(\\frac{j-s}{2}\\Big)\\prod_{m=1}^{M-1}\\Gamma(\\nu_m-s)\\,\n\\int_0^\\infty \\frac{dy}{y}\\,y^{\\nu_M-s}e^{-y}\n\\end{equation}\nand the lemma follows by induction.\n\\end{proof}\n\nAs already mentioned, functions with a contour integral representation like~\\eqref{contour-matrix} have certain properties which are useful for analytical purposes.\nIn fact, many of these properties may be found in the literature if we first recognise the contour integral as a Fox $H$-function\n\\begin{equation}\\label{weight-fox}\n g_j^{(M)}(x)=\\frac{(\\sgn x)^{j}}2\n\\FoxH{M+1}{0}{0}{M+1}{-}{(\\nu_1,1),\\ldots,(\\nu_M,1),(\\frac{j}2,\\frac12)}{|x|}\n\\end{equation}\nor as a Meijer $G$-function\n\\begin{equation}\\label{weight}\n g_j^{(M)}(x)=(\\sgn x)^{j}\\prod_{m=1}^M\\frac{2^{\\nu_m-1}}{\\sqrt{\\pi}}\n \\MeijerG{2M+1}{0}{0}{2M+1}{-}{\\frac{\\nu_1}{2},\\frac{\\nu_1+1}{2},\\ldots,\\frac{\\nu_M}{2},\\frac{\\nu_M+1}{2},\\frac{j}2}{\\frac{x^2}{4^M}}.\n\\end{equation}\nWe refer to the book~\\cite{MSH09} for an extensive review of these functions; the Fox $H$- and Meijer $G$-functions are defined by~\\cite[Def.~1.1]{MSH09} and~\\cite[Def.~1.5]{MSH09}, respectively.\n\nAs discussed in Section~\\ref{sec:motivation}, one of our goals is to find a `classical gas' approximation for~\\eqref{PDF-matrix}. For this purpose, we can use the asymptotic result~\\cite{Fi72}\n\\begin{equation}\\label{asymp}\n\\MeijerG{q}{0}{0}{q}{-}{b_1,\\ldots,b_q}{x}\\sim\\frac1{q^{1\/2}}\\Big(\\frac{2\\pi}{x^{1\/q}}\\Big)^{(q-1)\/2}\nx^{(b_1+\\cdots+b_q)\/{q}}e^{-qx^{1\/q}}\\big(1+O(x^{1\/q})\\big),\n\\end{equation}\nfor $x\\to\\infty$ (recall that a typical eigenvalue grows with $n$). This immediately allows us to find a Muttalib--Borodin ensemble~\\eqref{MB},\nwhich approximates the product ensemble~\\eqref{PDF-matrix}.\nHowever, for notational simplicity, it is convenient to first make a change of variables\n\\begin{equation}\\label{change}\nx_i\\mapsto x_i'=2^M\\Big(\\frac{y_i}{\\sqrt{2M+1}}\\Big)^{2M+1}\n\\end{equation}\nfor $i=1,\\ldots,n$. Using the asymptotic formula~\\eqref{asymp} and making a change of variable~\\eqref{change}, we find the approximate PDF\n\\begin{equation}\\label{hermite-MB}\n\\tilde P^{(M)}(y_1,\\ldots,y_n)=\\frac{1}{\\tilde Z^{(M)}}\\prod_{1\\leq i<j\\leq n} (y_j-y_i)(y_j^{2M+1}-y_i^{2M+1})\n\\prod_{k=1}^n|y_k|^\\alpha e^{-y_k^2},\n\\end{equation}\nwhere $\\tilde Z^{(M)}$ is a new normalisation constant and $\\alpha=\\sum_{m=1}^M(2\\nu_m+1)$.\nWe recognise~\\eqref{hermite-MB} as the Hermite Muttalib--Borodin ensemble~\\eqref{MB-Hermite}.\nNote that the approximation~\\eqref{asymp} is valid for large $x$ and that the absolute value of a typical eigenvalue grows with the matrix dimension $n$. Thus, one might suspect agreement between the two models in the large-$n$ limit except for local correlations near the origin. We will return to a comparison between the two models in Section~\\ref{sec:hard} and Section~\\ref{sec:global}.\n\n\nIt is worth noting that the following exact relation holds\n\\begin{equation}\\label{asymp-exact}\n\\MeijerG{q}{0}{0}{q}{-}{0,\\frac1q,\\ldots,\\frac{q-1}q}{x}=\\frac{(2\\pi)^{(q-1)\/2}}{q^{1\/2}}e^{-qx^{1\/q}}\n\\end{equation}\nfor integer $q$.\nThis may be proven by writing the Meijer $G$-function on the left-hand side as its integral representation and then using Gauss' multiplication formula for the gamma functions. The exact relation~\\eqref{asymp-exact} tells us that we can choose the parameters $b_k$ $(k=1,\\ldots,q)$ in~\\eqref{asymp} such that all subleading terms in the expansion vanish, and (\\ref{hermite-MB}) is exact.\n\n\n\\section{Biorthogonality and correlations}\n\\label{sec:biortho}\n\nGenerally polynomial ensembles  describe determinantal point processes. The correlation kernel may be written as\n\\begin{equation}\\label{kernel-sum}\nK_n(x,y)=\\sum_{k=0}^{n-1}\\frac{p_k(x)\\phi_k(x)}{h_k},\n\\end{equation}\nwhere $p_k(x)$ and $\\phi_k(x)$ are bi-orthogonal functions with normalisation $h_k$, i.e.\n\\begin{equation}\n\\int_{-\\infty}^\\infty dx\\, p_k(x)\\phi_k(x)=h_k\\delta_{k\\ell}.\n\\end{equation}\nFor both \\eqref{PDF-matrix} and \\eqref{hermite-MB} the $p_{k}(x)$ will be a monic polynomial of degree $k$, while\n\\begin{equation}\n\\phi_k(x)-g_k^{(M)}(x)\\in\\Span\\{g_{k-1}^{(M)}(x),\\ldots,g_{0}^{(M)}(x)\\}\n\\end{equation}\nfor the product ensemble~\\eqref{PDF-matrix} and\n\\begin{equation}\n\\phi_k(x)-x^{(2M+1)k} |x|^{\\alpha}e^{-x^2}\\in\\Span\\{x^{(2M+1)(k-1)} |x|^{\\alpha}e^{-x^2},\\ldots, |x|^{\\alpha}e^{-x^2}\\}\n\\end{equation}\nfor the Hermite Muttalib--Borodin ensemble~\\eqref{hermite-MB}. In the latter case, the bi-orthogonal structure is already known~\\cite{Ko67,Ca68,Bo98,FI16}. The main purpose of this section is to determine the bi-orthogonal functions --- and consequently the kernel~\\eqref{kernel-sum} --- for the product ensemble~\\eqref{PDF-matrix}.\n\n\\subsection{The oddness of being even}\n\nFor $M=0$, the bi-orthogonal functions are $p_n(x)=\\tilde H_n(x)$ and $\\phi_n(x)=\\tilde H_n(x)e^{-x^2}$ with $\\tilde H_n(x)$ denoting the Hermite polynomials in monic normalisation. We recall that the $n$-th Hermite polynomials is an even (odd) function when $n$ is even (odd);\nthis is due to the reflection symmetry of the Gaussian weight about the origin. A similar phenomenon is present for our product generalisation. In order to see this, we use an alternative form of the kernel~\\eqref{kernel-sum}. We have~\\cite{Co39,Bo98}\n\\begin{equation}\\label{kernel}\nK_n(x,y)=\\sum_{k,\\ell=0}^{n-1}(B_{n}^{(M)})^{-1}_{\\ell,k}\\,x^{k}g_{\\ell}^{(M)}(y),\n\\end{equation}\nwhere $B_n^{(M)}=(b_{i,j}^{(M)})_{i,j=0}^{n-1}$ is the $n$-th bi-moment matrix constructed from the bi-moments\n\\begin{equation}\\label{bimoment}\nb_{k,\\ell}^{(M)}=\\int_{-\\infty}^\\infty x^kg_\\ell^{(M)}(x)dx.\n\\end{equation}\nSimon~\\cite{Si08} refers  the inverse moment matrix representation~\\eqref{kernel} as the ABC (Aitken--Berg--Collar) theorem.\n\nIn the following, it will be useful to extend the concept of odd and even moments to bi-moments.\nWe say that the bi-moments are odd (even) when $k+\\ell$ is odd (even).\nNow, using that $g_\\ell^{(M)}(-x)=(-1)^\\ell g_\\ell^{(M)}(x)$,\nwe see that the bi-moments satisfy\n\\begin{equation}\n b_{k,\\ell}^{(M)}=(-1)^{k+\\ell}b_{k,\\ell}^{(M)},\\qquad k,\\ell=0,1,\\ldots,\n\\end{equation}\nwhich implies that all odd moments are equal to zero. In other words, the entries in the bi-moment matrix vanishes in a chequerboard pattern. Thus, by reordering rows and columns we may write the bi-moment matrix in a block diagonal form\n$B_n^{(M)}\\mapsto \\diag(B_n^{(M,\\text{even})},B_n^{(M,\\text{odd})})$ with $B_n^{(M,\\text{even})}=(b_{2k,2\\ell}^{(M)})_{k,\\ell}$ and\n$B_n^{(M,\\text{odd})}=(b_{2k+1,2\\ell+1}^{(M)})_{k,\\ell}$.\nUsing this reordering in the sum~\\eqref{kernel}, we see that the kernel splits into two parts\n\\begin{equation}\nK_n(x,y)=K_n^\\text{even}(x,y)+K_n^\\text{odd}(x,y),\n\\end{equation}\nwhere\n\\begin{align}\nK_n^\\text{even}(x,y)&=\\sum_{k,\\ell=0}^{\\lfloor \\frac{n-1}2\\rfloor}(B_n^{(M,\\text{even})})^{-1}_{\\ell,k}\\,x^{2k}g_{2\\ell}^{(M)}(y)\n=\\sum_{k=0}^{\\lfloor \\frac{n-1}2\\rfloor} \\frac{p_{2k}(x)\\phi_{2k}(x)}{h_{2k}}, \\label{kernel-even} \\\\\nK_n^\\text{odd}(x,y)&=\\sum_{k,\\ell=0}^{\\lfloor \\frac n2\\rfloor-1}(B_n^{(M,\\text{odd})})^{-1}_{\\ell,k}\\,x^{2k+1}g_{2\\ell+1}^{(M)}(y)\n=\\sum_{k=0}^{\\lfloor \\frac n2\\rfloor-1} \\frac{p_{2k+1}(x)\\phi_{2k+1}(x)}{h_{2k+1}}. \\label{kernel-odd}\n\\end{align}\nHere, the latter equality in both~\\eqref{kernel-even} and~\\eqref{kernel-odd} follow from comparison with~\\eqref{kernel-sum}.\nFinally, we note that\n\\begin{equation}\nK_{2n}^\\text{even}(x,y)=K_{2n-1}^\\text{even}(x,y)\n\\qquad\\text{and}\\qquad\nK_{2n}^\\text{odd}(x,y)=K_{2n+1}^\\text{odd}(x,y).\n\\end{equation}\nThus, in the following we can restrict our attention to kernels with an even subscript.\n\n\n\\subsection{Bi-orthogonal functions}\n\nWe are now ready to write down the bi-orthogonal functions for our product ensemble. As explained in the previous subsection, it is convenient to consider functions of odd and even degree separately.\n\\begin{prop}\\label{thm:bi-func-sum}\nThe ensemble defined by Theorem~\\ref{cor-matrix} is bi-orthogonalised by\n\\begin{align}\np_{2n}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\n\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+1)}{\\Gamma(\\nu_m+2\\ell+1)}x^{2\\ell}, &\n\\phi_{2n}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n)!}{(2\\ell)!}g_{2\\ell}^{(M)}(x), \\nonumber \\\\\np_{2n+1}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\n\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+2)}{\\Gamma(\\nu_m+2\\ell+2)}x^{2\\ell+1}, &\n\\phi_{2n+1}(x)&=\\sum_{\\ell=0}^n\\frac{(-\\tfrac14)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n+1)!}{(2\\ell+1)!}g_{2\\ell+1}^{(M)}(x), \\nonumber \\\\\nh_n&=2^{-n}\\pi^{1\/2}\\prod_{m=0}^M\\Gamma(\\nu_m+n+1)\\label{h_n},\n\\end{align}\nwith notation as above (recall that $\\nu_0=0$).\n\\end{prop}\n\nThere are several different approaches to prove Proposition~\\ref{thm:bi-func-sum}. Here, we will present a method which emphasizes the relation to the Hermite polynomials (see~\\cite[Prop.~3.5]{Ip15} for the same method applied to the product ensemble of Laguerre type). In order to use this approach, it is convenient to first calculate the bi-moments.\n\n\\begin{lemma}\nThe bi-moments are given by\n\\begin{equation}\\label{bimoment-gamma}\nb_{k,\\ell}^{(M)}=\\Gamma\\Big(\\frac{k+\\ell+1}{2}\\Big)\\prod_{m=1}^M\\Gamma(\\nu_m+k+1),\n\\end{equation}\nfor $k+\\ell$ even and zero otherwise.\nThe bi-moment determinant is\n\\begin{equation}\\label{bi-det}\nD_n^{(M)}:=\\det_{0\\leq k,\\ell\\leq n}[b_{k\\ell}^{(M)}]=2^{-n(n+1)\/2}\\pi^{(n+1)\/2}\\prod_{m=0}^M\\prod_{j=0}^n\\Gamma(\\nu_m+j+1).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe insert the contour integral representation of the weight functions~\\eqref{contour-matrix} into the expression for the bi-moments~\\eqref{bimoment}, then we see that the even moments are\n\\begin{equation}\nb_{k\\ell}=\\frac{1}{2\\pi i}\\int_0^\\infty dx\\int_{c-i\\infty}^{c+i\\infty} ds \\,x^s\\,\\Gamma\\Big(\\frac{k+\\ell-s}{2}\\Big)\n\\prod_{m=1}^M\\Gamma(\\nu_m+k-s).\n\\end{equation}\nThe integrals in this expression can be recognised as a combination of a Mellin and an inverse Mellin transform, which yields~\\eqref{bimoment-gamma}.\n\nIn order to evaluate the bi-moment determinant~\\eqref{bi-det}, we note that\n\\begin{equation}\\label{bimoment-det}\nD_n^{(M)}=\\prod_{m=1}^M\\prod_{j=0}^n\\Gamma(\\nu_m+j+1)\\det_{0\\leq k,\\ell\\leq n}[b_{k\\ell}^{(M=0)}].\n\\end{equation}\nThis completes the proof, since the $M=0$ case is the well-known Hermite (or GUE) case.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{thm:bi-func-sum}]\nThe bi-orthogonal functions may be expressed by means of their bi-moments exactly as orthogonal polynomials through their moments.\nThus, we have\n\\begin{equation}\np_n(x)=\\frac{1}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M)} \\bigg\\vert\\, x^i\n\\end{bmatrix}\n\\qquad\\text{and}\\qquad\n\\phi_n(x)=\\frac{1}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n-1\\\\j=0,\\ldots,n}}\n\\bigg[\\begin{array}{@{}cc @{}}\nb_{i,j}^{(M)}  \\\\ \\hline\ng_j^{(M)}(x)\n\\end{array}\\bigg].\n\\end{equation}\nFurthermore, we have $h_n=D_n^{(M)}\/D^{(M)}_{n-1}$.\n\nThe constants $h_n$ are immediate from the above and~\\eqref{bi-det}. Thus, it remains only to find the bi-orthogonal functions. To do so, we first note that\n\\begin{align}\np_n(x)&=\\frac{\\prod_{k=0}^n\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M=0)} \\bigg\\vert\\, \\displaystyle\\frac{x^i}{\\prod_{m=1}^M\\Gamma(\\nu_m+i+1)}\n\\end{bmatrix},\\\\\n\\phi_n(x)&=\\frac{\\prod_{k=0}^{n-1}\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}{D_{n-1}^{(M)}}\n\\det_{\\substack{i=0,\\ldots,n-1\\\\j=0,\\ldots,n}}\n\\bigg[\\begin{array}{@{}cc @{}}\nb_{i,j}^{(M=0)}  \\\\ \\hline\ng_j^{(M)}(x)\n\\end{array}\\bigg].\n\\end{align}\nThis observation is important, since we know that the monic Hermite polynomials (with respect to the weight $e^{-x^2}$) are given by\n\\begin{equation}\n\\tilde H_n(x)=2^{-n}H_n(x)=\n\\frac{1}{D_{n-1}^{(M=0)}}\n\\det_{\\substack{i=0,\\ldots,n\\\\j=0,\\ldots,n-1}}\n\\begin{bmatrix}\nb_{i,j}^{(M=0)} \\bigg\\vert\\, x^i\n\\end{bmatrix}.\n\\end{equation}\nIt follows that the expressions for the bi-orthogonal function $p_n(x)$ and $\\phi_n(x)$ can be found using the known expressions for the Hermite polynomials and then making substitutions\n\\begin{equation}\nx^k\\mapsto \\frac{x^k}{\\prod_{m=1}^M\\Gamma(\\nu_m+k+1)}\n\\qquad\\text{and}\\qquad\nx^k\\mapsto g_k^{(M)}(x),\n\\end{equation}\nrespectively. We recall that\n\\[\n\\tilde H_{2n}(x)=\\sum_{\\ell=0}^n\\frac{(-1)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n)!}{(2\\ell)!}\\frac{(2x)^{2\\ell}}{2^{2n}}\n\\quad\\text{and}\\quad\n\\tilde H_{2n+1}(x)=\\sum_{\\ell=0}^n\\frac{(-1)^{n-\\ell}}{(n-\\ell)!}\\frac{(2n+1)!}{(2\\ell+1)!}\\frac{(2x)^{2\\ell+1}}{2^{2n+1}},\n\\]\nwhich makes it a straightforward exercise to verify the proposition.\n\\end{proof}\n\n\\begin{remark}\nWe recall that the bi-orthogonal functions also can be obtained from the characteristic polynomial using that\n\\begin{equation}\np_N(x)=\\big\\langle\\det[x\\mathbb I_N-W_M]\\,\\big\\rangle\n\\qquad\\text{and}\\qquad\n\\int_{\\mathbb R} dx\\,\\frac{\\phi_{N-1}(x)}{z-x}=\\Big\\langle\\,\\frac1{\\det[z\\mathbb I_N-W_M]}\\,\\Big\\rangle\n\\end{equation}\nwith $\\langle\\cdots\\rangle$ denoting the matrix average and $z\\in\\mathbb C\\setminus\\mathbb R$. The first relation allows for an alternative method to calculate $p_N(x)$; see e.g.~\\cite{FL15}.\n\\end{remark}\n\n\n\\section{Integral representations and correlation kernels}\n\\label{sec:int}\n\nThe explicit expressions for the bi-orthogonal functions given by Proposition~\\ref{thm:bi-func-sum} allow us to write down an explicit form for the correlation kernel by insertion in~\\eqref{kernel-sum}. However, this formulation of the kernel is not optimal for asymptotic analysis. For this reason, in this section we will provide integral representations of the bi-orthogonal functions as well as the kernel.\n\n\\begin{prop}\\label{prop:bi-func-int}\nThe bi-orthogonal functions given by Proposition~\\ref{thm:bi-func-sum} have integral representations\n\\begin{align}\np_{2n}(|x|)&=\\frac{\\sqrt\\pi(2n)!}{(-1)^n2^{2n}}\\frac{1}{2\\pi i}\\oint_\\Sigma ds\\,|x|^{2s}\\,\n\\frac{\\Gamma(-s)}{\\Gamma(n+1-s)\\Gamma(s+\\frac12)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2n+1)}{\\Gamma(\\nu_m+2s+1)}, \\label{p2n-int} \\\\\np_{2n+1}(|x|)&=\\frac{\\sqrt\\pi(2n+1)!}{(-1)^n2^{2n+1}}\\frac{1}{2\\pi i}\\oint_\\Sigma ds\\,|x|^{2s+1}\\,\n\\frac{\\Gamma(-s)}{\\Gamma(n+1-s)\\Gamma(s+\\frac32)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2n+2)}{\\Gamma(\\nu_m+2s+2)}, \\label{p2n1-int} \\\\\n\\frac{\\phi_{2n}(|x|)}{h_{2n}}&=\\frac{(-1)^n\\,2^{2n}}{\\sqrt\\pi(2n)!}\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} dt\\,\n|x|^{-2t-1}\\frac{\\Gamma(n-t)\\Gamma(t+\\frac12)}{\\Gamma(-t)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2n+1)}, \\\\\n\\frac{\\phi_{2n+1}(|x|)}{h_{2n+1}}&=\\frac{(-1)^n\\,2^{2n+1}}{\\sqrt\\pi(2n+1)!}\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} dt\\,\n|x|^{-2t-2}\\frac{\\Gamma(n-t)\\Gamma(t+\\frac32)}{\\Gamma(-t)}\n\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+2)}{\\Gamma(\\nu_m+2n+2)},\n\\end{align}\nwhere the contour $\\Sigma$ encloses the integers $0,1,\\ldots,n$ in the negative direction and $-\\frac12<c<0$.\nWe recall that $p_{2n}(x)$ and $\\phi_{2n}(x)$ are even functions, while $p_{2n+1}(x)$ and $\\phi_{2n+1}(x)$ are odd functions.\n\\end{prop}\n\n\\begin{proof}\nThe integrands in~\\eqref{p2n-int} and~\\eqref{p2n1-int} has $n+1$ simple poles located at $0,1,\\ldots,n$, thus the series representations in Proposition~\\ref{thm:bi-func-sum} follow upon a straightforward application of the residue theorem.\n\nIn order to find the integral representation of the bi-orthogonal functions $\\phi_n(x)$, we first note that the weight functions can be written as\n\\begin{equation}\\label{weight-int-proof}\ng_\\ell^{(M)}(x)=(\\sgn x)^\\ell\\int_0^\\infty \\frac{dy}{y}\\,y^\\ell\\,e^{-y^2}\n\\MeijerG{M}{0}{0}{M}{-}{\\nu_1,\\ldots,\\nu_M}{\\frac{|x|}{y}};\n\\end{equation}\nthis is easily seen starting from the recursive definition~\\eqref{weight-recursive}. Now, using~\\eqref{weight-int-proof} in the expression for $\\phi_n(x)$ (cf. Proposition~\\ref{thm:bi-func-sum}), we see that\n\\begin{equation}\n\\phi_n(x)=\\int_0^\\infty \\frac{dy}{y}e^{-y^2}\\tilde H_n(y)\n\\MeijerG{M}{0}{0}{M}{-}{\\nu_1,\\ldots,\\nu_M}{\\frac{|x|}{y}}.\n\\end{equation}\nIn other words, the bi-orthogonal functions $\\phi_n(x)$ are an integral transform of the Hermite polynomials with respect to a Meijer $G$-function as integral kernel. The Hermite polynomial can itself be expressed as a Meijer $G$- or Fox $H$-function (see~\\cite[Sec.~1.8.1.]{MSH09}) and the remaining integral is well-known from the literature~\\cite[Sec.~2.3.]{MSH09}.\n\\end{proof}\n\n\n\\begin{prop}\\label{prop:kernel-finite}\nIntegral representations of the even and odd kernels are\n\\begin{align}\n K_{2n}^\\textup{even}(x,y)&=\\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t}\n \\frac{\\Gamma(-\\frac s2)\\Gamma(\\frac{t+1}2)}{\\Gamma(-\\frac t2)\\Gamma(\\frac{s+1}2)}\\frac{\\Gamma(\\frac {2n-t}2)}{\\Gamma(\\frac{2n-s}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)}, \\nonumber \\\\\n K_{2n}^\\textup{odd}(x,y)&=\\frac{\\sgn(xy)}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{s}\\,|y|^{-t-1}}{s-t}\\,\n \\frac{\\Gamma(\\frac{1-s}2)\\Gamma(\\frac{t+2}2)}{\\Gamma(\\frac{1-t}2)\\Gamma(\\frac{s+2}2)}\n \\frac{\\Gamma(\\frac{2n-t+1}2)}{\\Gamma(\\frac{2n-s+1}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)},\n\\end{align}\nwith $-1<c<0$ and the contour $\\Sigma$ is chosen such that it encircles $0,1,\\ldots,2n-1$ in the negative direction with $\\textup{Re}\\{s\\}>c$ for all $s\\in\\Sigma$.\n\\end{prop}\n\n\\begin{proof}\nAs the proofs for the odd and even kernels are almost identical, we provide only the proof for the even case. The odd case is easily verified by the reader.\n\nIt follows the definition of the even kernel~\\eqref{kernel-even} together with contour integral representation of the bi-orthogonal functions from Proposition~\\ref{prop:bi-func-int}, that\n\\begin{equation}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int dt\\oint_\\Sigma ds\\,|x|^{2s}|y|^{-2t-1}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}\n \\sum_{k=0}^{n-1}\\frac{\\Gamma(k-t)}{\\Gamma(k+1-s)}.\n\\end{equation}\nFollowing similar steps as in~\\cite{KZ14}, we note the sum allows a telescopic evaluation. This gives\n\\begin{multline}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\frac{\\Gamma(n-t)}{\\Gamma(n-s)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}\\\\\n -\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(t+\\frac12)}{\\Gamma(s+\\frac12)}\\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}.\n\\end{multline}\nHere, the integrand on the second line is zero as it has no poles encircled by the contour $\\Sigma$ and, thus\n\\begin{equation}\n K_{2n}^\\text{even}(x,y)=\\frac{1}{(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\oint_\\Sigma ds\\,\\frac{|x|^{2s}|y|^{-2t-1}}{s-t}\n \\frac{\\Gamma(-s)\\Gamma(t+\\frac12)}{\\Gamma(-t)\\Gamma(s+\\frac12)}\\frac{\\Gamma(n-t)}{\\Gamma(n-s)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+2t+1)}{\\Gamma(\\nu_m+2s+1)}.\n\\end{equation}\nFinally, the proposition follows by a change of variables $s\\mapsto s\/2$ and $t\\mapsto t\/2$.\n\\end{proof}\n\n\nThe above integral representations for the bi-orthogonal functions and the kernel are probably the most convenient form for further asymptotic analysis, as we will see in Section~\\ref{sec:hard}. However, it is also often helpful to express these formulae in terms of special functions as (for example) it allows for use of pre-defined mathematical software. Furthermore, such reformulations often guide us to recognise patterns which otherwise would have been left unseen.\n\nThe integral representations for the bi-orthogonal functions given by Proposition~\\ref{prop:bi-func-int} can also be recognised as several different types of special function; this includes generalised hypergeometric, Meijer $G$-, and Fox $H$-functions. Here, we will restrict ourselves to their Meijer $G$-function formulation.\n\nLet us first consider the bi-orthogonal polynomials which may be written as\n\\begin{align}\np_{2n}(x)&=\\frac{(-1)^n}{2^{2n}}\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+1)}{2^{\\,\\nu_m}\\pi^{-1\/2}}  \\nonumber\n\\MeijerG{1}{0}{1}{2M+2}{n+1}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2+\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2+\\frac12}{\\frac{x^2}{2^{2M}}}, \\\\\n\\frac{p_{2n+1}(x)}x&=\\frac{(-1)^n}{2^{2n}}\\prod_{m=0}^M\\frac{\\Gamma(\\nu_m+2n+2)}{2^{\\,\\nu_m+1}\\pi^{-1\/2}}\n\\MeijerG{1}{0}{1}{2M+2}{n+1}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2-\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2-\\frac12}{\\frac{x^2}{2^{2M}}}.\n\\label{p_2n-meijer}\n\\end{align}\nIt is worth comparing these polynomials with the polynomial found in the study of the Laguerre-like matrix product~\\eqref{old-product}.\nAkemann et al.~\\cite{AIK13} found that in this case the bi-orthogonal polynomial is given by\n\\begin{equation}\nP_n^{(M)}(x)=(-1)^n\\prod_{m=0}^M\\Gamma(\\nu_m+n+1)\n\\MeijerG{1}{0}{0}{M+1}{n+1}{-\\nu_0,-\\nu_1,\\ldots,-\\nu_M}{x}.\n\\end{equation}\nIt is clear that the two families of polynomials are related as\n\\begin{equation}\np_{2n}(x)\\propto P_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\qquad\\text{and}\\qquad\np_{2n+1}(x)\\propto xP_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\end{equation}\nwith\n\\begin{equation}\\label{nu-map}\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}-1)\/2\\}_{m=0}^M\n\\qquad\\text{and}\\qquad\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}+1)\/2\\}_{m=0}^M,\n\\end{equation}\nrespectively.\nThis is a generalisation of the relation between Hermite and Laguerre polynomials. Recall that\n\\begin{equation}\\label{HL}\n\\tilde H_{2n}(x)=\\tilde L^{(-\\frac12)}_n(x^2)\n\\qquad\\text{and}\\qquad\n\\tilde H_{2n+1}(x)=x\\tilde L^{(+\\frac12)}_n(x^2),\n\\end{equation}\nwhere $\\tilde H_n(x)$ and $\\tilde L_n^{(\\alpha)}(x)$ denote the Hermite and Laguerre polynomials in monic normalisation.\n\nLikewise, the (non-polynomial) bi-orthogonal functions may be written as\n\\begin{align}\n\\frac{\\phi_{2n}(|x|)}{|x|}&=(-1)^n\\prod_{m=1}^M\\frac{2^{\\nu_m-2}}{\\pi^{1\/2}}\n\\MeijerG{2M+1}{1}{1}{2M+2}{-n}{\\frac{\\nu_M}2-\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2-\\frac12,\\frac{\\nu_0}2}{\\frac{x^2}{2^{2M}}}, \\\\\n\\phi_{2n+1}(|x|)&=(-1)^n\\prod_{m=1}^M\\frac{2^{\\nu_m-1}}{\\pi^{1\/2}}\n\\MeijerG{2M+1}{1}{1}{2M+2}{-n}{\\frac{\\nu_M}2+\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2+\\frac12,\\frac{\\nu_0}2}{\\frac{x^2}{2^{2M}}}.\n\\end{align}\nAgain, we want to compare to the formula in~\\cite{AIK13} which this time reads\n\\begin{equation}\n\\Phi_n^{(M)}(x)=(-1)^n\\MeijerG{M}{1}{1}{M+1}{-n}{\\nu_M,\\ldots,\\nu_1,\\nu_0}{x}.\n\\end{equation}\nEvidently, we have the following relations\n\\begin{equation}\\label{phi-relations}\n\\phi_{2n}(|x|)\\propto |x|\\Phi_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big)\n\\qquad\\text{and}\\qquad\n\\phi_{2n+1}(|x|)\\propto \\Phi_n^{(2M+1)}\\Big(\\frac{x^2}{2^{2M}}\\Big),\n\\end{equation}\nwith~\\eqref{nu-map} as before.\nYet again, this is a generalisation of the relation between Hermite and Laguerre polynomials. In the simplest case the relations~\\eqref{phi-relations} reduces to\n\\begin{equation}\n\\tilde H_{2n}(x)w_\\text{H}(x)=|x|\\tilde L^{(-\\frac12)}_n(x^2)w_\\text{L}^{(-\\frac12)}(x^2)\n\\qquad\\text{and}\\qquad\n\\tilde H_{2n+1}(|x|)w_\\text{H}(x)=\\tilde L^{(+\\frac12)}_n(x^2)w_\\text{L}^{(\\frac12)}(x^2), \\nonumber\n\\end{equation}\nwhere $w_\\text{H}(x)=e^{-x^2}$ and $w_\\text{L}^{(\\alpha)}(x)=x^\\alpha e^{-x}$ are the Hermite and Laguerre weight functions.\n\nIt is, of course, well-known that there are relations between ensembles with reflection symmetry about the origin and ensembles on the half-line (albeit explicit formulae may be elusive). A general description of such relations in the Muttalib--Borodin ensembles can be found in~\\cite{FI16}.\n\n\n\n\n\\section{Scaling limits at the origin in product and Muttalib--Borodin ensembles}\n\\label{sec:hard}\n\n\nWith the integral representations of the correlation kernels established by Proposition~\\ref{prop:kernel-finite}, we can turn to a study of asymptotic properties. Perhaps the most interesting scaling regime is that of the local correlations near the origin, referred to as the hard edge when the eigenvalues are strictly positive. For other product ensembles~\\cite{KZ14,Fo14,KKS15,FL16}, it has been observed that correlations at the hard edge is determined by the so-called Meijer $G$-kernel, which generalises the more familiar Bessel kernel. Below, we will see that the Meijer $G$-kernel  appears once again, but this time\ninvolving a sum.\n\n\\begin{thm}\\label{thm:hard} Let $K_n(x,y)=K_n^\\textup{even}(x,y)+K_n^\\textup{odd}(x,y)$ with the even and odd kernels given by Proposition~\\ref{prop:kernel-finite}. For $x,y\\in\\mathbb R\\setminus\\{0\\}$ and $\\nu_1\\ldots,\\nu_M$ fixed, the microscopic limit near the origin is\n\\begin{equation}\\label{hard-limit}\n\\lim_{n\\to\\infty}\\frac{1}{\\sqrt n}K_{2n}\\Big(\\frac x{\\sqrt n},\\frac y{\\sqrt n}\\Big)\n=K^\\textup{even}(x,y)+K^\\textup{odd}(x,y)\n\\end{equation}\nwith\n\\begin{align}\nK^\\textup{even}(x,y)\n&=\\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t}\n \\frac{\\Gamma(-\\frac s2)\\Gamma(\\frac{t+1}2)}{\\Gamma(-\\frac t2)\\Gamma(\\frac{s+1}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)} \\label{hard-even} \\\\\nK^\\textup{odd}(x,y)&=\\frac{\\sgn(xy)}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}\\,|y|^{-t-1}}{s-t}\\,\n \\frac{\\Gamma(\\frac{1-s}2)\\Gamma(\\frac{t+2}2)}{\\Gamma(\\frac{1-t}2)\\Gamma(\\frac{s+2}2)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)} \\label{hard-odd},\n\\end{align}\nwhere $-1<c<-1\/2$ and $\\Sigma$ encloses the non-negative half-line in the negative direction starting and ending at $+\\infty$ such that $\\textup{Re}\\{s\\}>c$ for all $s\\in\\Sigma$.\n\\end{thm}\n\n\\begin{proof}\nWe only consider the even kernel in Proposition~\\ref{prop:kernel-finite} since the odd case is very similar.  After rescaling we rewrite  the integral representation of the even kernel in Proposition~\\ref{prop:kernel-finite} as\n\\begin{align}\\label{5.4}\n\\frac 1{\\sqrt n} K_{2n}^\\textup{even}(\\frac x{\\sqrt n},\\frac y{\\sqrt n})= \\frac{1}{2(2\\pi i)^2}\\int_{c-i\\infty}^{c+i\\infty} dt\\int_\\Sigma ds\\,\\frac{|x|^{s}|y|^{-t-1}}{s-t} \\frac{f_{n}(s)}{f_{n}(t)} \\frac{g(t)}{g(s)},\n\\end{align}\n with \\begin{equation}\n f_{n}(s)= \\frac{\\Gamma(n)\\Gamma(-\\frac s2)}{n^{\\frac s2} \\Gamma(n-\\frac s2)}, \\qquad \n g(s)=\\Gamma\\big(\\frac{s+1}{2}\\big) \\prod_{m=1}^M \\Gamma(\\nu_m+s+1). \n \\end{equation}\n \n For any fixed $t\\in c+i \\mathbb{R}$  and $s \\in \\Sigma$, using \\cite[eq. 5.11.13]{NIST} we see \n\\begin{equation}\\label{5.6}\nf_{n}(s)= \\Gamma(-\\frac s2) \\big(1+O(\\frac{1}{n})\\big),  \\qquad f_{n}(t)= \\Gamma(-\\frac t2) \\big(1+O(\\frac{1}{n})\\big). \\end{equation}\nFormally, substituting (\\ref{5.6}) in (\\ref{5.4}) gives (\\ref{hard-even}). To proceed rigorously, we need to verify   a condition for the  exchange of limit and integration.  For this purpose, we  will proceed to \nfind two dominated functions respectively  corresponding to  $1\/|f_{n}(t)|$  and  $|f_{n}(s)|$.\n\nFirst,     using \\cite[eq. 5.11.13]{NIST} we have for sufficiently  large $n$\n\\begin{equation}\n \\frac{1}{|f_{n}(t)|} \\leq  \\frac{n^{\\frac c2} \\Gamma(n-\\frac c2) }{ \\Gamma(n)|\\Gamma(-\\frac t2)|} \\leq   \\frac{2}{|\\Gamma(-\\frac t2)|},   \\quad \\forall  t\\in c+i \\mathbb{R}. \\label{t-bound}\n\\end{equation}\n\nSecond, we require an upper bound  for $ | f_{n}(s)  |$. Noting the asymptotic expansion, that as $z\\rightarrow \\infty$ in the sector $|\\mathrm{arg}(z)|\\leq \\pi-\\delta$ (with $0<\\delta<\\pi$)\n\\begin{equation}\n\\Gamma(z)=e^{-z}z^{z-\\frac{1}{2}}\\sqrt{2\\pi} \\big(1+O(\\frac{1}{z})\\big), \\label{Agamma}\n\\end{equation} \nit is easy to see that for a given $y_0>0$ we can choose the contour $\\Sigma=\\Sigma_{l}\\cup \\Sigma_{r}$ with \\begin{equation} \\Sigma_{l}=\\big\\{\\frac{c}{2}+iy:|y|\\leq y_{0}\\big\\}\\cup \\big\\{x\\pm iy_0: \\frac{c}{2} \\leq x\\leq 1\\big\\}, \\quad \\Sigma_{r}=  \\big\\{x\\pm iy_0:   x> 1\\big\\}.\\end{equation}\nThus, we get from \\eqref{Agamma} and the boundedness of $\\Gamma(-s\/2)$ over $\\Sigma_l$ that  for large $n$ there exists a constant $C_1=C_1(y_0) > 0$ such that \n\\begin{equation}\n |f_{n}(s)| \\leq C_1, \\qquad \\forall s\\in  \\Sigma_{l}. \\label{upl}\n\\end{equation}\nIn order to estimate $f_{n}(s)$ with $s\\in  \\Sigma_{r}$, we use the integral representation  \n \\begin{equation}\n  f_{n}(s)=   \\frac{n^{-\\frac{s}{2}}}{ 2i \\sin \\frac{\\pi s}{2}}    \\int_{\\mathcal{C}_0} (1-u)^{n-1}(-u)^{-\\frac{s}{2}-1}du,\n\\end{equation}\nwhere  $\\mathcal{C}_0$ is a counter-clockwise path which begins and ends at $1$ and encircles the origin once; see e.g.  \\cite[eq. 5.12.10]{NIST}. Note that we choose $(-u)^{-1-s\/2}=e^{-(1+s\/2)\\log(-u)}$ with $-\\pi<\\mathrm{arg}(-u)<\\pi$. Change $u$ by $u\/n$ and deform the resulting contour into the path which starts from $n$, proceeds along the (upper) real axis to 1, describes a circle of radius one counter-clock round the origin and returns to $n$ along the (lower) real axis. That is, \n \\begin{equation}\n  f_{n}(s)=   \\frac{1}{ 2i \\sin \\frac{\\pi s}{2}}    \\int_{\\mathcal{C}} (1-\\frac{u}{n})^{n-1}(-u)^{-\\frac{s}{2}-1}du.\n\\end{equation}\nLet $s=v\\pm iy_0, v>1$. On the unit circle of the $u$-integral above  write $-u=e^{i\\theta}$. Then we easily obtain  for $n\\geq1$\n \\begin{equation}\n  |f_{n}(s)|\\leq    \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{-\\pi}^{\\pi} \\big(1+\\frac{1}{n}\\big)^{n-1} |e^{-(\\frac{s}{2}+1)i\\theta}|d\\theta\\leq \n  \\frac{\\pi e^{ 1+\\frac{\\pi y_{0}}{2}}}{   |\\sin \\frac{\\pi s}{2}|}. \\label{ub1}\n\\end{equation}\nOn the upper and lower  real axis,  we have \n \\begin{align}\n  |f_{n}(s)|&\\leq    \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{1}^{n} \\big(1-\\frac{u}{n}\\big)^{n-1}|u^{-\\frac{s}{2}-1} e^{-(\\frac{s}{2}+1)(\\mp i\\pi)}|du \\nonumber \\\\\n  &\\leq \\frac{1}{ 2 |\\sin \\frac{\\pi s}{2}|}    \\int_{1}^{n}   u^{-\\frac{v}{2}-1} e^{\\frac{1}{2}\\pi y_{0}} du \\nonumber\\\\\n & =\\frac{1}{  |\\sin \\frac{\\pi s}{2}|}  e^{\\frac{1}{2}\\pi y_{0}}    \\frac{1- n^{-\\frac{v}{2}} }{v}\\leq \\frac{1}{  |\\sin \\frac{\\pi s}{2}|}  e^{\\frac{1}{2}\\pi y_{0}}.  \\label{ub2}\n\\end{align} \nUsing the simple fact $|\\sin \\frac{\\pi s}{2}| \\geq |\\sinh \\frac{\\pi }{2} \\mathrm{Im}(s)|$, combination of \\eqref{ub1} and \\eqref{ub2} shows that \nthere exists a constant $C_2=C_2(y_0)>0$ such that \n\\begin{equation}\n |f_{n}(s)| \\leq C_2, \\qquad \\forall s\\in  \\Sigma_{r}.\n\\end{equation}\nTogether with \\eqref{upl} this gives us a bound $C>0$, that is, for large $n$  \n\\begin{equation}\n |f_{n}(s)| \\leq C, \\qquad \\forall s\\in  \\Sigma. \\label{s-bound}\n\\end{equation}\n\n\nFinally, use \\eqref{Agamma} and  the  asymptotic formula  that as $y\\rightarrow \\pm \\infty$ \n\\begin{equation}\n| \\Gamma(x + iy) |\\sim  \\sqrt{2\\pi} |y|^{x-\\frac{1}{2}}  e^{-\\frac{1}{2}\\pi |y|}\n\\end{equation}\nwith bounded real value of $x$ (see \\cite[eq. 5.11.9]{NIST}),  it is easy to conclude  that the  function  of  variables $s$ and $t$\n \\begin{equation}\n\n \\,\\frac{||x|^{s}|y|^{-t-1}|}{|s-t|}   \\frac{2}{|\\Gamma(-\\frac t2)|}   \\frac{|g(t)|}{|g(s)|},  \n \\end{equation}\n  is integrable along the chosen contours, \n whenever $-1<c<-1\/2$. Here  we emphasize that the assumption $-1\/2\\leq c<0$  does  not ensure the convergence in the special case $M=0$ while for  $M\\geq 1$ it can be relaxed to $-1<c<0$ as in Proposition~\\ref{prop:kernel-finite}.\n \nWith this, combing  \\eqref{t-bound} and \\eqref{s-bound}, we have indeed justified  the interchange of limit and integrals for every $M$ by the dominated convergence theorem, which completes the proof.\n\\end{proof}\n\n\n\nFor a comparison with other known results, it is useful to rewrite the hard edge correlation function of Theorem~\\ref{thm:hard} in terms of Meijer $G$-functions.\nUsing that\n\\begin{equation}\n\\int_0^1du\\,u^{s-t-1}=\\frac1{s-t},\n\\end{equation}\nwe see that the even and odd kernel can be written as\n\\begin{align}\nK^\\textup{even}(|x|,|y|)=\\frac{|y|}{2^{2M}}\\int_0^1du\\, \\nonumber\n&\\MeijerG{1}{0}{0}{2M+2}{-}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2+\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2+\\frac12}{\\frac{x^2}{2^{2M}}u} \\\\\n&\\times\\MeijerG{2M+1}{0}{0}{2M+2}{-}{\\frac{\\nu_M}2-\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2-\\frac12,\\frac{\\nu_0}2}{\\frac{y^2}{2^{2M}}u}, \\\\\nK^\\textup{odd}(|x|,|y|)=\\frac{|x|}{2^{2M}}\\int_0^1du\\, \\nonumber\n&\\MeijerG{1}{0}{0}{2M+2}{-}{-\\frac{\\nu_0}2,-\\frac{\\nu_0}2-\\frac12,\\ldots,-\\frac{\\nu_M}2,-\\frac{\\nu_M}2-\\frac12}{\\frac{x^2}{2^{2M}}u} \\\\\n&\\times\\MeijerG{2M+1}{0}{0}{2M+2}{-}{\\frac{\\nu_M}2+\\frac12,\\frac{\\nu_M}2,\\ldots,\\frac{\\nu_0}2+\\frac12,\\frac{\\nu_0}2}{\\frac{y^2}{2^{2M}}u},\n\\end{align}\nrespectively. We recall that the so-called Meijer $G$-kernel is given by~\\cite{KZ14}\n\\begin{equation}\nK_\\text{Meijer}^M(x,y)=\\int_0^1du\\,\\MeijerG{1}{0}{0}{M+1}{-}{-\\nu_0,\\ldots,-\\nu_M}{xu}\\MeijerG{M}{0}{0}{M+1}{-}{\\nu_M,\\ldots,\\nu_0}{yu} \\label{G-kernel}\n\\end{equation}\nwith $x,y>0$. We note that this kernel is single-sided ($x,y\\in\\mathbb R_+$) while the kernel from Theorem~\\ref{thm:hard} is double-sided ($x,y\\in\\mathbb R\\setminus\\{0\\}$). However, it is also evident that our new kernel may be re-expressed in terms of the Meijer $G$-kernel. We have\n\\begin{equation}\nK^\\textup{even}(|x|,|y|)=\\frac{|y|}{2^{2M}}K_\\text{Meijer}^{2M+1}\\Big(\\frac{x^2}{2^{2M}},\\frac{y^2}{2^{2M}}\\Big)\n\\quad\\text{and}\\quad\nK^\\textup{odd}(|x|,|y|)=\\frac{|x|}{2^{2M}}K_\\text{Meijer}^{2M+1}\\Big(\\frac{x^2}{2^{2M}},\\frac{y^2}{2^{2M}}\\Big)\n\\end{equation}\nwith\n\\begin{equation}\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}-1)\/2\\}_{m=0}^M\n\\qquad\\text{and}\\qquad\n\\{\\nu_m\\}_{m=0}^M\\mapsto\\{{\\nu_m}\/2,({\\nu_m}+1)\/2\\}_{m=0}^M,\n\\end{equation}\nrespectively. Thus, the random product matrix~\\eqref{W1} provides yet another appearance of the Meijer $G$-kernel; albeit this time in a double-sided version. For graphical representation of the Meijer $G$-kernel we refer to~\\cite[Fig.~3.2]{Ip15}, which shows plots of the local density (i.e. the kernel with $x=y$) for different values of $M$.\n\nA double-side hard edge scaling limit near the origin is also present in the Hermite Muttalib--Borodin ensemble.\nIn this case the kernel is found to be~\\cite{Bo98}\n\\begin{equation}\\label{kernel-borodin}\nK^\\text{even}(x,y)=K^{(\\frac{\\alpha-1}2,\\theta)}(x^2,y^2)\n\\qquad\\text{and}\\qquad\nK^\\text{odd}(x,y)=\\sgn(xy)|x|^\\theta|y|\\,K^{(\\frac{\\alpha+\\theta}2,\\theta)}(x^2,y^2)\n\\end{equation}\nwhere\n\\begin{equation}\\label{kernel-wright-bessel}\nK^{(\\alpha,\\theta)}(x,y)=\\theta\\int_0^1du(xu)^\\alpha J_{\\frac{\\alpha+1}\\theta,\\frac1\\theta}(xu)J_{\\alpha+1,\\theta}((yu)^\\theta)\n\\end{equation}\nwith $J_{a,b}(x)$ denoting Wright's Bessel function. In the case relevant to us~\\eqref{MB-Hermite}, we also have $\\theta=2M+1$. Furthermore, it is known from~\\cite{KS14} that the kernel~\\eqref{kernel-wright-bessel} is a Meijer $G$-kernel whenever $\\theta$ is a positive integer. In particular, we have\n\\begin{equation}\n\\Big(\\frac{x^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}-1}\nK^{(\\alpha,2M+1)}\\Big((2M+1)\\Big(\\frac{x^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}},(2M+1)\\Big(\\frac{y^2}{2^{2M}}\\Big)^{\\frac{1}{2M+1}}\\Big)\n=K_\\text{Meijer}^{2M+1}\\Big(\\frac{y^2}{2^{2M}},\\frac{x^2}{2^{2M}}\\Big),\n\\end{equation}\nwhere the Meijer $G$-kernel on the right-hand side has indices\n\\begin{equation}\n\\nu_m=\\frac{\\alpha+m-1}{2M+1},\\qquad m=1,\\ldots,2M+1,\n\\end{equation}\nand as always $\\nu_0=0$. It follows from~\\eqref{kernel-borodin} and~\\eqref{kernel-wright-bessel} that the hard edge correlations for the Hermite Muttalib--Borodin ensemble with appropriately chosen parameters may be expressed in terms of the Meijer $G$-kernel in a similar fashion as done for the product ensemble above. We note that the choice of variables in~\\eqref{kernel-wright-bessel} should be compared to the change of variables~\\eqref{change} performed in the derivation of the asymptotic reduction~\\eqref{hermite-MB}.\n\n\nIt is worth verifying consistency of the simplest scenario of $M=0$.\nWhen  $M=0$ our matrix ensemble~\\eqref{W1} reduces to the GUE, hence the kernel given by Theorem~\\ref{thm:hard} must reduce to the sine kernel for $M=0$. To see this, we use \n\\begin{equation}\n\\MeijerG{1}{0}{0}{2}{-}{0,\\frac12}{\\frac{x^2}{4}}=\\frac{\\cos x}{\\sqrt\\pi}\n\\qquad\\text{and}\\qquad\n\\MeijerG{1}{0}{0}{2}{-}{\\frac12,0}{\\frac{x^2}{4}}=\\frac{\\sin |x|}{\\sqrt\\pi}.\n\\end{equation}\nIt follows that\n\\begin{align}\nK^\\textup{even}(x,y)&=\\frac{1}{\\pi}\\int_0^1\\frac{du}{\\sqrt u}\\cos(2x\\sqrt u)\\cos(2y\\sqrt u)\n=\\frac1{\\pi}\\Big(\\frac{\\sin 2(x-y)}{2(x-y)}+\\frac{\\sin 2(x+y)}{2(x+y)}\\Big), \\\\\nK^\\textup{odd}(x,y)&=\\frac{1}{\\pi}\\int_0^1\\frac{du}{\\sqrt u}\\sin(2x\\sqrt u)\\,\\sin(2y\\sqrt u)\n=\\frac1{\\pi}\\Big(\\frac{\\sin 2(x-y)}{2(x-y)}-\\frac{\\sin 2(x+y)}{2(x+y)}\\Big),\n\\end{align}\nwhich upon insertion into~\\eqref{hard-limit} indeed reproduces the sine kernel.\n\nIn the end of this section, let us emphasize that there  also exists a contour integral  representation of the limiting kernel in Theorem~\\ref{thm:hard} which combines the odd and even into a single formula.\n\n\\begin{prop}\\label{prop:kernelrep2} With the same  notation as  in Theorem \\ref{thm:hard}, the limiting kernel at the origin can be rewritten as\n\\begin{align}\n K^\\textup{even}(x,y)+ K^\\textup{odd}(x,y)=2\\, \\mathcal{K}_{\\nu_{1},\\ldots,\\nu_{M}}(2x,2y), \\label{equivalence}\n \\end{align}\nwhere the kernel on the right-hand side is defined as\n\\begin{align}\n\\mathcal{K}_{\\nu_{1},\\ldots,\\nu_{M}}(x,y)&\n=\\int_{C_{R}} \\frac{dv}{2\\pi i}\n\\,\\MeijerG{1}{0}{0}{M+1}{-}{0,-\\nu_1, \\ldots,-\\nu_M}{-\\sgn(y)xv}\\MeijerG{M+1}{0}{0}{M+1}{-}{0, \\nu_1,\\ldots,\\nu_M}{|y|v},\n\\label{doubleG-kernel}\n\\end{align}\nwith $C_{R}$ denoting a path in the right-half plane from $-i$ to $i$.\n\\end{prop}\n\n\\begin{proof}\nUsing  Euler's reflection formula and duplication formula for the gamma function, we see that\n\\begin{equation*}\nK^\\textup{even}(x,y)+ K^\\textup{odd}(x,y)=\\frac{1}{(2\\pi i)^2}\\int dt\\int ds\\, (2|x|)^{s}(2|y|)^{-t-1}\n\\frac{ g(s,t)}{s-t} \\frac{\\Gamma(t+1)}{\\Gamma(s+1)}\n \\prod_{m=1}^M\\frac{\\Gamma(\\nu_m+t+1)}{\\Gamma(\\nu_m+s+1)},\n\\end{equation*}\nwhere\n\\begin{equation}\ng(s,t)=\\frac{\\sin\\frac{\\pi}{2}t}{\\sin\\frac{\\pi}{2}s}+\\sgn(xy) \\frac{\\cos\\frac{\\pi}{2}t}{\\cos\\frac{\\pi}{2}s}.\n\\end{equation}\nIn order to proceed, we will consider the cases $xy<0$ and $xy>0$ separately.\nFor $xy<0$,  it is seen that\n\\begin{equation}\ng(s,t)= \\frac{2}{\\sin\\pi s}\\sin\\frac{\\pi}{2}(t-s)=-\\frac{2}{\\pi} \\Gamma(-s)\\Gamma(1+s) \\, \\sin\\frac{\\pi}{2}(t-s).\n\\end{equation}\nNow~\\eqref{equivalence} can be obtained using the integral representation\n\\begin{equation}\n \\frac{1}{\\pi i} \\int_{C_{R}}dv \\,v^{s-t-1}= \\frac{1}{t-s}\\sin\\frac{\\pi}{2}(t-s),\n\\end{equation}\nwith the contour $C_R$ as above,\ntogether with the definition of Meijer $G$-function. For $xy>0$, we note that\n\\begin{equation}\ne^{i\\pi s}g(s,t)=\n\\left(\\frac{\\sin\\frac{\\pi}{2}t}{\\sin\\frac{\\pi}{2}s}-\\frac{\\cos\\frac{\\pi}{2}t}{\\cos\\frac{\\pi}{2}s} \\right)+2e^{i\\frac{\\pi}{2}(t+s)}.\n\\end{equation}\nThe $s$-variable integrand in the second part has no pole within the contour $\\Sigma$. Thus, the problem reduces to the proven situation.\n\\end{proof}\n\nThe simplest non-trivial case is $M=1$. Here, we get\n \\begin{equation}\n\\mathcal{K}_{\\nu}(x,y)\n=\\left(\\frac{y}{x}\\right)^{\\nu\/2}  \\frac{1}{\\pi i} \\int_{C_{R}}dv\n\\, I_{\\nu}\\big(2\\sqrt{\\sgn(y)xv}\\big)\\,  K_{\\nu}\\big(2\\sqrt{|y|v}\\big),  \\label{doubleM1-kernel}\n\\end{equation}\nwith the modified Bessel functions $I_{\\nu}$ and $K_{\\nu}$, which follows immediately from the fact that\n \\begin{align}\n\\MeijerG{1}{0}{0}{2}{-}{0,-\\nu}{-z}=z^{-\\nu\/2} I_{\\nu}(2\\sqrt{z}), \\qquad  \\MeijerG{2}{0}{0}{2}{-}{\\nu,0}{z}=2 z^{\\nu\/2} K_{\\nu}(2\\sqrt{z}).\n\\end{align}\n\n\n\n\\section{Global spectra in product and Muttalib--Borodin ensembles}\n\\label{sec:global}\n\nThe study of the scaling limit  at  the origin  in the previous section introduces a scale in which the average spacing between eigenvalues is of order unity. A very different, but still well-defined, limiting process is the so-called global scaling regime. In this regime the average spacing between eigenvalues tends to zero in such way that the spectral density tends to a quantity $\\rho(x)$ with compact support $I\\subset\\mathbb R$ and $\\int_I \\rho(x)dx=1$. Here $\\rho(x)$ is referred to as the global density.\nThroughout this section, the indices $\\nu_1\\ldots,\\nu_M$ are kept fixed.\n\nFor the Laguerre Muttalib--Borodin ensemble specified by the density~\\eqref{MB-laguerre} the global scaling limit corresponds to a change of variables $x_j\\mapsto nx_j$. Introducing the further change of variables $x_j\\mapsto Mx_j^M$, the global density is known to be the so-called Fuss--Catalan density with parameter $M$ \\cite{FW15}. It can be specified by the moment sequence\n\\begin{equation}\n\\text{FC}_M(k)=\\frac1{Mk+1}\\binom{(M+1)k}k,\\qquad k=0,1,\\ldots\\,.\n\\end{equation}\nThese are the Fuss--Catalan numbers (the Catalan numbers are the case $M=1$).\n\nNow, consider the product of $M$ standard complex Gaussian random matrices. Consistent with the discussion in Section~\\ref{sec:motivation}, the corresponding global density is again the Fuss--Catalan density with parameter $M$~\\cite{Mu02,AGT10,BBCC11,NS06}.\n\nIt is known that the Fuss--Catalan density, $\\rho^{(M)}_\\text{FC}(x)$ say, can also be characterised as the minimiser of the energy functional\n\\begin{equation}\\label{energy-laguerre}\nE[\\rho]=M\\int_0^Ldx\\,\\rho(x)x^{\\frac1M}-\\frac{1}{2}\\int_0^Ldx\\int_0^Ldy\\,\\rho(x)\\rho(y)\\log\\big(|x-y||x^{\\frac1M}-y^{\\frac1M}|\\big)\n\\end{equation}\nwith $L=(M+1)^{M+1}\/M^M$; see~\\cite{CR14,FL15,FLZ15}. Note that the energy functional~\\eqref{energy-laguerre} relates to~\\eqref{MB-laguerre} through the aforementioned change of variables. Similarly, the energy functional corresponding to~\\eqref{MB-Hermite} is\n\\begin{align}\n\\tilde E[\\tilde\\rho]&=\\theta\\int_{-\\tilde L}^{\\tilde L}dx\\,\\rho(x)x^{\\frac2\\theta}\n-\\frac{1}{2}\\int_{-\\tilde L}^{\\tilde L}dx\\int_{-\\tilde L}^{\\tilde L}dy\\,\n\\tilde\\rho(x)\\tilde\\rho(y)\\log\\big(|x-y||\\sgn x|x|^{\\frac1\\theta}-\\sgn y|y|^{\\frac1\\theta}|\\big) \\nonumber \\\\\n&=2\\theta\\int_{0}^{\\tilde L}dx\\,\\rho(x)x^{\\frac2\\theta}-\\int_{0}^{\\tilde L}dx\\int_{0}^{\\tilde L}dy\\,\n\\tilde\\rho(x)\\tilde\\rho(y)\\log\\big(|x^2-y^2||(x^2)^{\\frac1\\theta}-(y^2)^{\\frac1\\theta}|\\big)\n\\label{energy-hermite}\n\\end{align}\nwith $\\theta=2M+1$.\nWe note that changing variables $x^2\\mapsto x$ and $y^2\\mapsto y$, then setting $\\tilde\\rho(x)=x\\rho(x^2)$ reduces~\\eqref{energy-hermite} to~\\eqref{energy-laguerre} with $L=\\tilde L^2$. Thus, the minimiser in~\\eqref{energy-hermite} is given in terms of the Fuss-Catalan density\n\\begin{equation}\\label{double-sided-FC}\n\\tilde\\rho(x)=|x|\\rho_\\text{FC}^{(M)}(x^2)\n\\end{equation}\nand is symmetric about the origin.\n\nAs an illustration, let us consider the simplest case, $M=1$. The Fuss--Catalan density becomes the celebrated Mar\\v cenko--Pastur density,\n\\begin{equation}\n\\rho_\\text{FC}^{(M=1)}(x)=\\frac{1}{2\\pi}\\sqrt{\\frac{4-x}{x}},\\qquad 0<x<4.\n\\end{equation}\nThe formula~\\eqref{double-sided-FC} then gives the standard result (see e.g.~\\cite{PSbook}) that the energy functional\n\\begin{equation}\n \\tilde E[\\tilde\\rho]=\\int_{-2}^{2}dx\\,\\tilde \\rho(x)x^{2}\n-\\int_{-2}^{2}dx\\int_{-2}^{2}dy\\,\\tilde\\rho(x)\\tilde\\rho(y)\\log|x-y|\n\\end{equation}\nis minimised by\n\\begin{equation}\n\\rho_\\text{Wigner}(x)=\\frac{\\sqrt{4-x^2}}{2\\pi},\\qquad -2<x<2,\n\\end{equation}\nwhich is Wigner's semi-circle law.\n\nIt has been demonstrated in Section~\\ref{sec:product}, that the energy function implicit in~\\eqref{energy-hermite} underlies the eigenvalue distribution of the random matrix product~\\eqref{W1}. Thus, we can anticipate that after appropriate scaling the global density for the product ensembles is given by~\\eqref{double-sided-FC}. A direct proof of this can obtained through a number of different strategies. We consider first a method based on the characteristic polynomial.\n\nIn terms of the global scaled variables, the key equation relating the averaged characteristic polynomial to  the global is the asymptotic formula~\\cite{FW15}\n\\begin{equation}\\label{asymp-stieltjes}\n\\frac1n\\frac d{dz}\\log\\big\\langle\\det(z n^{M+\\frac12}\\mathbb I_n-W_M)\\big\\rangle=\\tilde G_M(z)+O(n^{-1}),\n\\end{equation}\nwhere $\\tilde G_M(z)$ is the Stieltjes transform of the global spectral density,\n\\begin{equation}\\label{stieltjes}\n\\tilde G_M(z)=\\int_{-\\tilde L}^{\\tilde L}dx\\,\\frac{\\tilde\\rho(x)}{z-x}.\n\\end{equation}\nFollowing the strategy first used in~\\cite{FL15}, the formula~\\eqref{asymp-stieltjes} leads to a characterisation of the Stieltjes transform~\\eqref{stieltjes}, upon realising that the characteristic polynomial satisfy a linear differential equation.\n\n\\begin{prop}\nConsider a matrix product~\\eqref{W1} with even matrix dimension, $n=2N$. Let\n\\begin{equation}\\label{char-poly}\nf(z)=\\big\\langle\\det(z\\mathbb I_{n}-W_M)\\big\\rangle\n\\end{equation}\ndenote the characteristic polynomial. Then~\\eqref{char-poly} is a solution to the $(2M+2)$-th differential equation,\n\\begin{equation}\\label{diff}\n2z^2\\Big(z\\frac d{dz}-n\\Big)f(z)\n=\\prod_{m=0}^M\\Big(z\\frac d{dz}+\\nu_m\\Big)\\Big(z\\frac d{dz}+\\nu_m-1\\Big)f(z),\n\\end{equation}\nwith asymptotic boundary condition $f(z)\\sim z^n$ for $|z|\\to \\infty$.\n\\end{prop}\n\n\\begin{proof}\nThe characteristic polynomial~\\eqref{char-poly} is identical to the bi-orthogonal polynomial $p_{2N}(z)$. As shown earlier, this polynomial is proportional to a Meijer $G$-function~\\eqref{p_2n-meijer}. It is well-known that such Meijer $G$-functions satisfy the differential equation~\\eqref{diff}. The asymptotic boundary condition follows trivially, since $f(z)$ is a monic polynomial.\n\\end{proof}\n\nChanging variables $z\\mapsto n^{M+\\frac12}\\hat z\/\\sqrt{2}$ in~\\eqref{diff} and using that~\\cite{FL15}\n\\begin{equation}\n\\frac{f^{(k)}(\\hat z)}{f(\\hat z)}\\sim\\Big(\\frac{f'(\\hat z)}{f(\\hat z)}\\Big)^k\n\\end{equation}\nto leading order in $n$, we see that for large $n$ the differential equation~\\eqref{diff} reduces to the algebraic equation (see e.g. \\cite{bai07} for $M=1$)\n\\begin{equation}\\label{alg-eq1}\nz^2(z\\tilde G_M(z)-1)=(z\\tilde G_M(z))^{2M+2}\n\\end{equation}\nwith asymptotic condition $\\tilde G_M(z)\\sim 1\/z$ as $|z|\\to\\infty$.\nThis equation is to be compared to the algebraic equation satisfied by the Stieltjes transform of the Fuss--Catalan density,\n\\begin{equation}\\label{alg-eq2}\nz(zG_M(z)-1)=(zG_M(z))^{M+1},\n\\end{equation}\nsee e.g.~\\cite{FL15}. With $z\\mapsto z^2$ and $M\\mapsto 2M+1$ and setting $\\tilde G_{M}(z)=zG_{2M+1}(z^2)$, we see that~\\eqref{alg-eq2} reduces to~\\eqref{alg-eq1}. This prescription is equivalent to ~\\eqref{double-sided-FC}, thus verifying this formula as the evaluation of the global density.\n\n\nThe same result can also be obtained using free probability techniques. To see this, we need some additional notation. Let $a$ be a non-commutative random variable with distribution $d\\mu(x)=\\rho(x)dx$. The Stieltjes transform $G_a(z)$ of the variable $a$ is defined analogous to~\\eqref{stieltjes}. The $S$-transform is defined as\n\\begin{equation}\nS_a(z)=\\frac{1+z}{z}\\gamma^{-1}(z)\n\\qquad\\text{with}\\qquad\n\\gamma(z)=-1+z^{-1}G_a(z^{-1}).\n\\end{equation}\nNow assume that $a$ and $b$ are two freely independent non-commutative random variables and that the Stieltjes transform $G_b(z)$ satisfies a functional equation $P(z,G_b(z))=0$.\nIt is known~\\cite{NS06} that under these conditions the Stieltjes transform $G_{ab}(z)$ of the product $ab$ satisfies\n\\begin{equation}\\label{functional-recursion}\nP\\Big(zS_a(zG_{ab}(z)-1),\\frac{zG_{ab}(z)}{S_a(zG_{ab}(z)-1)}\\Big)=0.\n\\end{equation}\nMoreover, we know that if $a$ is given by the free normal distribution (i.e. Wigner's semi-circle) and $b$ is given by the free Poisson distribution (i.e. Mar\\v cenko--Pastur), then\n\\begin{equation}\nS_a(z)=\\frac1{1+z} \\qquad\\text{and}\\qquad G_{b}(z)^2-zG_b(z)+1=0.\n\\end{equation}\nWe can now use that the limiting distributions for the GUE and the Wishart ensemble are the free normal and the free Poisson, respectively. Thus, using~\\eqref{functional-recursion} $M$ times, we see that our product~\\eqref{W1} indeed gives rise to the the functional equation~\\eqref{alg-eq1}.\n\nIt is also possible to construct a parametrisation of the global density in terms of elementary functions based on the polynomial equation \\eqref{alg-eq1}. With\n\\begin{equation}\nx_{0}^{2}=\\frac{\\big(\\sin((2M+2)\\varphi)\\big)^{2M+2}}{\\sin\\varphi\\,\\big(\\sin((2M+1)\\varphi)\\big)^{2M+1}},\n\\qquad  0\\leq\\varphi\\leq\\frac{\\pi}{2M+2},\n\\end{equation}\nwe have\n\\begin{equation}\n\\tilde\\rho(x_0\n=\\frac{1}{\\pi}\\sqrt{\\frac{\\sin\\varphi}{\\sin(2M+ 1)\\varphi}}\\left(\\frac{\\sin(2r+1)\\varphi}{\\sin(2M+ 2)\\varphi}\\right)^{ M}\\sin\\varphi,\n\\qquad  0\\leq\\varphi\\leq\\frac{\\pi}{2M+2};\n\\end{equation}\nsee e.g.~\\cite{FL15}.  We remark that it follows that the singularity at the origin blows up like\n\\begin{equation}\n\\tilde\\rho(x_0)\\sim  \\frac{1}{\\pi } \\sin\\frac{\\pi}{2M+2} \\, |x_0|^{-\\frac{M}{M+1}}\n\\end{equation}\nas $x_0\\to0$.\n\n\n\\section{Conclusion and outlook}\n\nIn this paper, we have shown that it is possible to construct a Hermitised random matrix product for which the eigenvalues form a determinantal point process  on the entire real line with an explicit kernel. This is a fundamental new contribution to the study of random matrix product ensembles, since all previous exactly solvable models of this type have had eigenvalues restricted to the positive half-line. Furthermore, we have argued that this Hermitised product ensemble can be considered a natural generalisation of the classical Hermite ensemble (i.e. GUE) in similar way as the squared singular values of matrix products with  Gaussian matrices~\\cite{AKW13,AIK13} and truncated unitary matrices~\\cite{KKS15} can be considered generalisations of the Laguerre and Jacobi ensembles, respectively.\nTo this point, we have shown that the joint eigenvalue PDF reduces asymptotically to the Muttalib--Borodin ensemble of Hermite type.\n\nOn another front, we have shown that the local scaling limit near the origin is described by a two-sided generalisation of the so-called Meijer $G$-kernel~\\cite{KZ14}. This two-sided kernel reduces to the sine kernel in the simplest case. We have also seen that the global density can be found explicitly and that it is expressed in terms of the so-called Fuss--Catalan distribution in a simple manner. Our result relies on an explicit double contour integral formulation of the correlation kernel (Proposition \\ref{prop:kernel-finite}).\nIt is worth stressing that we could make full use of this double contour integral formulation and give an analytical proof of the global density.  In fact, following almost exactly the same steps introduced in  \\cite{LWZ14}, it can be proven that the sine-kernel arises in the bulk\nand that Airy-kernel arises at the soft edge; cf. the proof of Theorems 1.1 and Theorem 1.3 as well as Remark 2 in~\\cite{LWZ14}. The full details are beyond the scope of this paper, so let us only mention that a basic starting  point is to approximate the integrand by elementary functions and rewrite the kernel,  say the even part, as\n\\begin{multline}\n\\frac{1}{n \\tilde\\rho(x_0)}\\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1}K_{2n}^\\text{even}\n\\bigg(\n\\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1} \\Big(\\frac{x_0}{\\sqrt{2}}+\\frac{x}{ \\tilde\\rho(x_0)n}\\Big), \\Big( \\sqrt{\\frac{2}{n}} \\Big)^{2M+1}(\\frac{x_0}{\\sqrt{2}}+\\frac{x}{ \\tilde\\rho(x_0)n}\\Big)\\bigg)\n\\\\ \\sim \\frac{\\sqrt{2}}{|x_0|\\tilde\\rho(x_0)}\\frac{1}{(2\\pi i)^2}\\int dt\\int ds\\,\\frac{e^{n(g(s)-g(t))}}{s-t}  \\Big|1+\\frac{\\sqrt{2}x}{ x_0\\tilde\\rho(x_0)n}\\Big|^{2ns} \\Big|1+\\frac{\\sqrt{2}y}{ x_0\\tilde\\rho(x_0)n}\\Big|^{-2nt-1}\n \\frac{h_{n}(s)}{h_{n}(t)},\n\\end{multline}\nwhere the phase function is given by\n\\begin{equation}\ng(z)=(2M+1)z-2(M+1)z\\log z+z\\log(z-1)-\\log(1-z)+z\\log x_{0}^2.\n\\end{equation}\nHence, the saddle point equation $g'(z)=0$ is exactly  expressed through the equation  \\eqref{alg-eq1}.   We stress that the above  parametrisation representation  plays a key role in the proof of the sine-kernel  via the steepest decent method.\n\nFinally, we emphasize that our construction of a Hermitised random product ensemble is based on a matrix transformation which maps  the space of polynomial ensembles onto itself (Theorem~\\ref{T1}). This type of matrix transformation are important since they preserve exact solvability. Our  proof of Theorem~\\ref{T1} is applicable  to the Hermitised product ensemble multiplied by a Gaussian matrix, crucially with the help of the   hyperbolic HCIZ integral over the pseudo-unitary group. However, it would be interesting to see whether this could be extended to the product ensemble multiplied by   other types of random matrices, say, truncated unitary matrices. For this, a possible way is to first extend the matrix integral formula stated  in \\cite[Theorem 2.3]{KKS15} from  the unitary group to  the  pseudo-unitary case, and then perform  the same steps as in Section \\ref{sec:fyodorov}. This will be an interesting and  challenging problem for us.\n\n\\paragraph{Acknowledgements}\nWe thank S. Kumar  for  useful discussions on his paper and M. Kieburg for comments on a first draft.\nWe acknowledge support by the Australian Research Council through grant DP170102028 (PJF),\nthe ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF,JRI), and  partially by ERC Advanced Grant \\#338804,  the National Natural Science Foundation of China  \\#11771417,  the Youth Innovation Promotion Association CAS  \\#2017491, the Fundamental Research Funds for the Central Universities \\#WK0010450002,   Anhui Provincial Natural Science Foundation \\#1708085QA03 (DZL).   D.-Z. Liu is particularly grateful to L\\'{a}szl\\'{o} Erd\\H{o}s  for funding   his one-year stay at  IST Austria.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Sec_introduction}\n\nNonlinear responses have been giving rise to a lot of research interest in condensed matter physics. For instance, the nonlinear optical response provides a powerful tool for spectrometry. It has been used to obtain a real-space imaging of the parity-violating magnetic order in insulators~\\cite{Fiebig2005SHG_review,VanAken2007ToroidalDomain} and to explore exotic order in spin-orbit coupled metals and superconductors~\\cite{Petersen2006SHG,Zhao2015SHG_Sr2IrO4,zhao2017global,Harter2017SHG_Cd2Re2O7}. In the optical nonlinear responses, energy of irradiating light  is usually larger than that of electron bands, and the observed signals are attributed to interband transitions~\\cite{Sturman1992Book}. On the other hand, intraband transitions are also important in conductivity measurements in which the frequency of the probe is usually lower than that of optical probes and comparable to the electronic energy scale. It is therefore expected that nonlinear responses are informative for investigating metallic compounds where the intraband transitions are relevant. \n\nRegarding the nonlinear response in metals, the second-order nonlinear conductivity (NLC) measurement has attracted much attention. Previous studies are mainly divided into two streams; field-induced NLC and field-free NLC. The former can be traced back to Rikken's seminal works~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy}. They realized the longitudinal NLC under an external magnetic field, and significant enhancement has recently been discovered in strongly spin-orbit coupled semiconductors and superconductors~\\cite{Tokura2018nonreciprocal_review}. The microscopic origin of the longitudinal NLC is attributed to a semiclassical contribution which we call Drude term~\\cite{Ideue2017,Wakatsuki2017,Itahashi2020STOnonreciprocal}. On the other hand, a lot of theoretical and experimental efforts have recently been devoted to the transverse NLC, that is, nonlinear Hall effect~\\cite{Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2}. The nonlinear Hall effect realized without the magnetic field is rooted in a geometric quantity named Berry curvature dipole (BCD)~\\cite{Sodemann2015}.\n\nAccording to the symmetry argument, the second-order NLC requires violation of parity symmetry \\Pa{}. The condition is satisfied by the acentric property of crystals which were previously studied~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Tokura2018nonreciprocal_review,Ideue2017,Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2}. In contrast, the \\Pa{}-symmetry breaking can also be accompanied by the magnetic order, that is called odd-parity magnetic multipole order~\\cite{Spaldin2008a,Watanabe2018grouptheoretical,Hayami2018Classification}. It is expected that counterparts of the NLC exist in magnetic metals.\n\nA key to the odd-parity magnetic multipole order is locally-noncentrosymmetric property of crystals. With such structure of crystals, the local site-symmetry of atoms does not have \\Pa{}-symmetry although the global \\Pa{}-symmetry is preserved owing to the sublattice degree of freedom~\\cite{Maruyama2012,Yanase2014zigzag}. Then, the anti-symmetric spin-orbit coupling  (ASOC) emerges in a sublattice dependent way and gives rise to exotic responses such as the antiferromagnetic Edelstein effect~\\cite{Yanase2014zigzag,Zelezny2014NeelorbitTorque,Hayami2014h,hikaruwatanabe2017}. Supposing the antiferromagnetic order preserving the translational symmetry, both of the \\Pa{} and \\T{}-symmetries may be violated while the combined symmetry, namely \\PT{}-symmetry, is preserved. Such parity-violating but \\PT{}-symmetric magnetic order is called odd-parity magnetic multipole order and has been discussed in the context of multipole physics~\\cite{Watanabe2018grouptheoretical,Hayami2018Classification} and antiferromagnetic spintronics~\\cite{Jungwirth2016,Manchon2019spin-orbit-torque_review,Watanabe2018}. More than 100 candidate materials such as BaMn$_2$As$_2$ and EuMnBi$_2$ have been identified~\\cite{Watanabe2018grouptheoretical}.\n\nIn this work, the NLC in odd-parity magnetic multipole metals are investigated. We present a general symmetry classification of NLC based on a quantum mechanical calculation. Supported by the microscopic analysis, we clarify NLC characteristic of magnetic metals with and without external magnetic field. We find that the NLC at $H=0$ is a measure of the ASOC. Furthermore, we reveal two types of field-induced NLC; the nematicity-assisted dichroism, and Berry curvature dipole effect induced by what we call magnetic ASOC. These phenomena originate from locally-noncentrosymmetric crystal structures and magnetic order, and hence have striking difference from the NLC in noncentrosymmetric (non-magnetic) crystals. We show the correspondence between \\T{}-symmetric and \\PT{}-symmetric systems in Table~\\ref{Table_NLC_MagneticField}.  \n\n\t\t\\begin{table}[htbp]\n\t\t\\caption{Second-order NLC in \\T{}\/\\PT{}-symmetric systems with\/without magnetic field $\\bm{H}$. Dominant contributions such as the BCD term are shown. The boldfaced terms are studied in this work.}\n\t\t\\label{Table_NLC_MagneticField}\n\t\t\\centering\n\t\t\\begin{tabular}{l|cc}\n\t\t\t\t\t&\\T{}\t&\\PT{}\t \\\\ \\hline\n\t\t$\\bm{H}=0$\t&BCD\t\t\t&\\textbf{Drude}\t \\\\\n\t\t$\\bm{H}\\ne 0$\t&magnetic Drude\t&\\textbf{magnetic BCD}\t \n\t\t\\end{tabular}\n\t\t\\end{table}\n\n\\section{Quantum theory and symmetry analysis}\\label{Sec_quantumtheory_symmetryanalysis}\n\nA theoretical treatment of the second-order NLC has been established in Sipe and his coworkers' works~\\cite{Sipe1993,Aversa1995,Sipe2000secondorder} where the nonlinear response functions are derived from straightforward extension of the linear response theory~\\cite{Kubo1957}. A detailed calculation of the NLC is shown in Appendix~\\ref{App_Sec_derivation_NLC}. Here we only describe the outline of derivation.\n\nA spatially-uniform electric field $\\bm{E}(t)$ is introduced so as to be compatible with calculations based on the Bloch states in the length gauge\nframework~\\cite{Sipe1993,Aversa1995,Sipe2000secondorder,Ventura2017,Passos2018,Parker2019}. Using the density matrix formalism, we derive the second-order electric current in the frequency domain as \n\t\t\\begin{align}\n\t\tJ^{\\mu (2)} (\\omega) = \\sum_{\\nu\\lambda} &\\int \\frac{d\\omega_1d\\omega_2}{2\\pi}  \\sigma^{\\mu;\\nu\\lambda} (\\omega; \\omega_1,\\omega_2 ) \\notag \\\\\n\t\t\t&\\times E^\\nu \\left( \\omega_1  \\right)E^\\lambda \\left( \\omega_2  \\right) \\delta (\\omega-\\omega_1-\\omega_2).\n\t\t\\end{align}\nAssuming the clean limit where the relaxation time $\\tau \\rightarrow \\infty$ within the transport regime $\\omega\\tau \\ll 1$, the NLC is classified by the dependence on the phenomenological relaxation time $\\tau = \\gamma^{-1}$ as\n\t\t\\begin{equation}\n\t\t\\sigma^{\\mu;\\nu \\lambda} = \\sigma^{\\mu;\\nu \\lambda}_\\text{D} + \\sigma^{\\mu;\\nu \\lambda}_\\text{BCD} + \\sigma^{\\mu;\\nu \\lambda}_\\text{int},\n\t\t\\label{nonlinear_conductivity}\n\t\t\\end{equation}\nin which the indices $\\nu,~\\lambda$ of applied electric fields are symmetric. The first two components are obtained as\n\t\t\\begin{align}\n\t\t\\sigma^{\\mu;\\nu \\lambda}_\\text{D} \n\t\t\t\t&= - \\frac{e^3}{\\gamma^2}\\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_a \\partial_\\mu \\partial_\\nu \\partial_\\lambda  \\epsilon_{\\bm{k}a}   f(\\epsilon_{\\bm{k}a}), \\label{drude}\\\\\n\t\t\\sigma^{\\mu;\\nu \\lambda}_\\text{BCD} \n\t\t\t\t&= \\frac{e^3}{2\\gamma}\\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_{a} \\epsilon_{\\mu\\nu\\kappa}   f(\\epsilon_{\\bm{k}a}) \\partial_\\lambda \\Omega^\\kappa_a + \\left[ \\nu \\leftrightarrow \\lambda \\right],\\\\\n\t\t\t\t&= \\frac{e^3}{2\\gamma} \\epsilon_{\\mu\\nu\\kappa}   \\mathcal{D}^{\\,\\lambda\\kappa} + \\left[ \\nu \\leftrightarrow \\lambda \\right], \\label{BCD_term}\n\t\t\\end{align}\nwhich are the Drude~\\cite{Ideue2017} and BCD~\\cite{Sodemann2015,deyo2009semiclassical,Moore2010} terms, respectively. The index $a$ represents the band index. We introduced the electron charge $e>0$ and the BCD defined as\n\t\t\\begin{equation}\n\t\t\\mathcal{D}^{\\,\\mu\\nu} = \\int \\frac{d\\bm{k}}{\\left( 2\\pi \\right)^d} \\sum_{a} f(\\epsilon_{\\bm{k}a}) \\partial_\\mu \\Omega^\\nu_a.\n\t\t\\end{equation}\nThese two terms are finite only in the metal state and divergent in the clean limit since both are Fermi surface terms. The remaining term $\\sigma^{\\mu;\\nu \\lambda}_\\text{int}$ comes from interband transitions, and it is not divergent in the clean limit~\\cite{Gao2014}. This term, therefore, gives a negligible contribution to the NLC in good metals.\n\nIn the group-theoretical classification of quantum phases, the parity-violating phases are classified into odd-parity electric\/magnetic multipole phases where \\T{}\/\\PT{}-symmetry is preserved~\\cite{Watanabe2018grouptheoretical,Hayami2018Classification}. It is known that these preserved symmetries impose strong constraints on the response functions~\\cite{hikaruwatanabe2017,Zelezny2017} in addition to equilibrium properties of the systems~\\cite{cracknell2016magnetism}. Thus, the symmetry analysis enables us to classify the NLC allowed in either \\T{}-symmetric or \\PT{}-symmetric systems based on the relaxation time dependence. The result is shown in Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity}.\n\nIn \\T{}-symmetric systems, all the terms in NLC are scaled by odd-order $O (\\tau^{2n+1})$. Recalling the linear response theory, the scattering rate $\\gamma$ can be replaced by the adiabaticity parameter whose sign represents irreversibility due to external fields~\\cite{Kubo1957}. Thus, the NLC should be accompanied by a dissipative response. This is consistent with previous theories~\\cite{Morimoto2018NonreciprocalElectronCorrelation,Hamamoto2019RachetScaling}. In contrast to the familiar linear conductivity, the Drude term is prohibited because it is even-order with respect to $\\tau$. The leading order term is the BCD term for the transverse NLC.\n\n On the other hand, the \\T{}-symmetry is broken by the magnetic order in the parity-violating \\PT{}-symmetric systems which we focus on. Therefore, the relaxation time dependence is even-order $O(\\tau^{2n})$, and intrinsic contributions $O(\\tau^{0})$ are allowed. The leading order term is the Drude term $O(\\tau^{2})$. We will show that the Drude term is a measure of the hidden ASOC characteristic of locally-noncentrosymmetric systems. The BCD term is prohibited to be consistent with the fact that the Berry curvature itself disappears due to the \\PT{}-symmetry. Although the effect of the \\T{}-symmetry breaking in acentric systems has been discussed in previous works~\\cite{Gao2014,Sodemann2015,Nandy2019,Du2019}, our classification has clarified the contrasting role of \\T{} and \\PT{}-symmetries in NLC. Below we will see that the \\PT{}-symmetry gives a clear insight into the NLC.\n\nIn our classification, extrinsic contributions such as the side jump and skew scattering are not taken into account~\\cite{Nandy2019,Du2019,CXiao2019ModifiedSemiclassics,Du2020quantumTheoryofNHE}. We, however, note that the extrinsic contributions may be similarly classified by the symmetries. Indeed, for nonmagnetic impurities with $\\delta$-function potential, we show that while extrinsic terms are allowed in the \\T{}-symmetric systems~\\cite{Du2019}, they are strongly suppressed by the \\PT{}-symmetry (See Appendix~\\ref{App_Sec_extrinsic}). This suppression is highly contrasting to the fact that the impurities play an important role in the NLC in \\T{}-symmetric materials such as WTe$_2$~\\cite{Kang2019}. When we focus on the \\PT{}-symmetric magnetic systems, the classification in Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity} is meaningful beyond the relaxation time approximation for impurity scattering.\n \n\t\t\\begin{table}[htbp]\n\t\t\\caption{Relaxation time dependence of the second-order NLC in \\T{}\/\\PT{}-symmetric systems. `N\/A' denotes that the component is forbidden by symmetry.}\n\t\t\\label{Table_relaxation_time_dependence_2nd_conductivity}\n\t\t\\centering\n\t\t$\n\t\t\t\t\\begin{array}{c|ccc}\n\t\t\t\t&\\sigma_\\text{D}\t&\\sigma_\\text{BCD}\t&\\sigma_\\text{int}\t\\\\ \\hline\n\t\t\t\t\\text{\\T{}}\t&\\text{N\/A}&O(\\tau)&O(\\tau^{-1})\\\\\n\t\t\t\t\\text{\\PT{}}&O(\\tau^2) &\\text{N\/A} &O(\\tau^{0})\n\t\t\t\t\\end{array}\n\t\t$\n\t\t\\end{table}\n\n\nAll the terms in NLC are allowed in the absence of both \\T{} and \\PT{}-symmetry. For instance, the Drude term becomes finite when we apply magnetic fields to originally \\T{}-symmetric systems~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Tokura2018nonreciprocal_review,Ideue2017}, that is described as `magnetic Drude' in Table~\\ref{Table_NLC_MagneticField}. Similarly, we expect a magnetic-field-induced NLC in originally \\PT{}-symmetric systems; the BCD term indeed arises from the \\PT{}-symmetry breaking (called `magnetic BCD' in Table~\\ref{Table_NLC_MagneticField}). This term is clarified in this work below. In the following, we consider the \\PT{}-preserving antiferromagnetic metal with or without the magnetic field, and discuss the Drude and BCD terms which are dominant in clean metals.\n\n\\section{NLC in odd-parity magnetic multipole systems}\\label{Sec_NLC_odd-parity_magnetic_multipole}\n\nWe introduce a minimal model of \\bma{} which undergoes odd-parity magnetic multipole order~\\cite{hikaruwatanabe2017}. Many magnetic compounds in the list of Ref.~\\cite{Watanabe2018grouptheoretical} belong to the same class. The Hamiltonian reads\n\t\t\\begin{equation}\n\t\t\tH(\\bm{k}) =\n\t\t\t\t\t\\epsilon(\\bm{k}) \\, \\tau_0+ \\bm{g} \\left( \\bm{k} \\right) \\cdot \\bm{\\sigma} \\, \\tau_z +\\bm{h} \\cdot \\bm{\\sigma} \\, \\tau_0 +  V_{\\rm AB} (\\bm{k}) \\, \\tau_x, \\label{BMA_model_Hamiltonian}\n\t\t\\end{equation}\nwhere $\\bm{\\sigma}$ and $\\bm{\\tau}$ are Pauli matrices representing the spin and sublattice degrees of freedom, respectively. In addition to the intra-sublattice and inter-sublattice hoppings, $\\epsilon (\\bm{k})$ and $V_\\text{AB}  (\\bm{k})$, we introduce the staggered $g$-vector $\\bm{g}  (\\bm{k})= \\bm{g}_0 (\\bm{k}) + \\bm{h}_\\text{AF}$ consisting of the sublattice-dependent ASOC $\\bm{g}_0  (\\bm{k})$~\\cite{Yanase2014zigzag,Zelezny2014NeelorbitTorque,Hayami2014h} and the molecular field $\\bm{h}_\\text{AF}= h_\\text{AF} \\hat{z}$ due to antiferromagnetic order in \\bma{} ~\\cite{Singh2009BaMn2As2_1,Singh2009BaMn2As2_2,Ramsal2013BMA_MagneticStructure}. The detailed material property of \\bma{} and expressions of $\\epsilon (\\bm{k})$, $V_\\text{AB}  (\\bm{k})$, and $\\bm{g}_0 (\\bm{k})$ are available in Appendix~\\ref{App_Sec_model_hamiltonian}. We also consider an external magnetic field $\\bm{h}$ to discuss field-induced NLC. \n\n\\subsection{Field-free nonlinear Hall effect}\\label{Sec_nonlinear_Hall_no_field}\n\nFirst, we show the NLC at zero magnetic field ($\\bm{h} =\\bm{0}$). Then, the NLC is mainly given by the Drude term (see Table~\\ref{Table_relaxation_time_dependence_2nd_conductivity}), and it is determined by the anti-symmetric and anharmonic property of the energy dispersion [see Eq.~\\eqref{drude}]. Such dispersion is known to be a pronounced property of the odd-parity magnetic multipole systems~\\cite{Yanase2014zigzag,Hayami2014h,Sumita2016,hikaruwatanabe2017,Watanabe2018grouptheoretical}. In the case of \\bma{}, the anti-symmetric component was identified to be a cubic term $k_xk_yk_z$~\\cite{hikaruwatanabe2017}. Indeed, the energy spectrum of the model Eq.~\\eqref{BMA_model_Hamiltonian} is obtained as\n\t\t\\begin{equation}\n\t\tE^\\pm_{\\bm{k}}= \\epsilon (\\bm{k}) \\pm \\sqrt{V_{\\rm AB}(\\bm{k})^2 + \\bm{g}(\\bm{k})^2 }. \\label{energyspectrum_no_magnetic_field}\n\t\t\\end{equation}\nThe anti-symmetric distortion in the band structure arises from the coupling term $\\bm{g}_0 (\\bm{k}) \\cdot \\bm{h}_\\text{AF}$ which is approximated by $\\sim k_xk_yk_z$ near time-reversal-invariant momentum. Thus, $\\sigma^{z;xy}$ and its cyclic components of NLC tensor are allowed. This indicates the nonlinear Hall effect, namely, the second-order electric current $J^z$ generated from the electric field $\\bm{E} \\parallel [110]$. For the strong antiferromagnet, $|\\bm{h}_\\text{AF}| \\gg |\\epsilon (\\bm{k})|$, $|V_\\text{AB}  (\\bm{k})|$, $|\\bm{g}_0 (\\bm{k})|$, the Drude component is analytically obtained as \n \t\t\\begin{equation}\n\t\t\\sigma^{z;xy}_\\text{D} =\\sigma^{x;yz}_\\text{D} =\\sigma^{y;zx}_\\text{D} = \\frac{e^3\\alpha_{\\parallel}n }{4\\gamma^2} \\,\\text{sgn\\,} (h_\\text{AF}), \\label{Drude_no_external_field}\n\t\t\\end{equation}\nin the lightly-hole-doped region. Here $n$ denotes the carrier density of holes and $\\alpha_{\\parallel}$ represents the strength of ASOC parallel to the staggered magnetization $\\bm{h}_\\text{AF}$. It is noteworthy that Eq.~\\eqref{Drude_no_external_field} does not depend on the antiferromagnetic molecular field and therefore it is useful to evaluate the \\textit{sublattice-dependent} ASOC. Thus, the NLC provides a way to experimentally deduce the sublattice-dependent ASOC, although it was called \"hidden spin polarization\"~\\cite{Zhang2014HiddenSpin,Gotlieb2018HiddenSpinInCuprate} because it is hard to be measured. Equivalence of $\\sigma^{z;xy}_\\text{D} =\\sigma^{x;yz}_\\text{D} =\\sigma^{y;zx}_\\text{D}$ holds independent of parameters and it can be tested by experiments. Numerical calculations of Eq.~\\eqref{drude} are consistent with the above-mentioned symmetry argument and analytic formula as shown in Appendix~\\ref{App_Sec_NLC_no_magnetic_field}. A typical value of the nonlinear Hall response is obtained as $\\sigma^{z;xy}_\\text{D}\/[(\\sigma^{xx})^2 \\sigma^{zz}] \\sim \\mr{10^{-17}}{[A^{-2} \\cdot V \\cdot m^3]}$ and it is much larger than the experimental value of bilayer WTe$_2$, $\\sigma^{y;xx}\/(\\sigma^{xx})^3 \\sim \\mr{10^{-19}}{[A^{-2} \\cdot V \\cdot m^3]}$~\\cite{Ma2019BCD_experiment_WTe2}. Because the Drude term is more divergent with respect to $\\tau$ than the BCD term, we may see a giant nonlinear Hall response in the \\PT{}-symmetric antiferromagnet.\n\nThe NLC is a useful quantity not only to evaluate the sublattice-dependent ASOC but also to detect domain states in antiferromagnetic metals~\\cite{Watanabe2018grouptheoretical}. Indeed, the sign of the NLC depends on the antiferromagnetic domain and hence it may promote developments in the antiferromagnetic spintronics~\\cite{Jungwirth2016,Manchon2019spin-orbit-torque_review}. In fact, the read-out of antiferromagnetic domains has been successfully demonstrated by making use of the NLC~\\cite{Godinho2018AFM_reading}. For \\bma{} and related materials listed in Ref.~\\cite{Watanabe2018grouptheoretical}, the nonlinear Hall effect can be used to identify antiferromagnetic domain states. So far we considered intrinsic contributions. We have shown that the extrinsic contributions from impurity scattering are suppressed due to the  preserved \\PT{}-symmetry, and therefore, they are not relevant to the above discussions.\n\n\\subsection{Nematicity-assisted dichroism}\\label{Sec_nematicity_assisted}\n\nIn the absence of the external field, \\bma{}-type magnetic materials do not show the longitudinal NLC along the high symmetry axes, namely, $\\sigma^{\\mu;\\mu\\mu}=0$. Below, we show that the longitudinal NLC can be induced by magnetic fields. Since the BCD term contributes to only the transverse response, we have only to consider the Drude term. Generally speaking, to obtain a finite longitudinal electronic dichroism, the system is required to possess an anti-symmetric dispersion such as $k_\\mu^3$ or higher-order one. According to the group-theoretical classification, the `polarization' in the momentum-space denoted by $k_\\mu$ may share the same symmetry as $k_\\mu^3$~\\cite{hikaruwatanabe2017,Watanabe2018grouptheoretical}. Thus, the momentum-space polarization is a key to realize the longitudinal dichroism.\n\nIn \\bma{} and related materials, the momentum-space polarization can be induced by the nematicity. We can understand this by the discussion of the magnetopiezoelectric effect~\\cite{Varjas2016,hikaruwatanabe2017,Shiomi2019EuMnBi2_MPE,Shiomi2019CaMn2Bi2_MPE,Shiomi2020}. A magnetopiezoelectric effect means that the planer (electronic) nematicity is induced by the out-of-plane electric current. That is written as\n\t\t\\begin{equation}\n\t\t\\varepsilon^{xy} = e^{xy;z} J^z,\n\t\t\\end{equation}\nwhere $\\varepsilon^{\\mu\\nu}$ represents the strain tensor.  It was experimentally discovered in EuMnBi$_2$~\\cite{Shiomi2019EuMnBi2_MPE,Shiomi2020} and CaMn$_2$Bi$_2$~\\cite{Shiomi2019CaMn2Bi2_MPE} in accordance with theoretical prediction. The response is derived from the anti-symmetrically distorted Fermi surface and hence realizable in the odd-parity magnetic multipole systems. Similar to the conventional piezoelectric effect, we may expect an inverse effect. Given the in-plane nematic order or strain, the system should obtain the momentum-space polarization $P^{\\,k_z}$ whose symmetry is the same as the electric current $J^z$,\n\t\t\\begin{equation}\n\t\tP^{\\,k_z} = \\tilde{e}^{z;xy} \\varepsilon^{xy}.\n\t\t\\end{equation}\n Accordingly, the longitudinal dichroism $\\sigma^{z;zz}$ is allowed. Thus, nematicity-assisted dichroism which is unique to the odd-parity magnetic multipole systems is implied. \n\nThe nematicity can be induced by the magnetic field through the spin-orbit coupling. In the model for \\bma{} the sublattice-dependent ASOC plays an essential role. By $\\bm{h} \\ne 0$, the energy spectrum of the lower bands $E_{\\bm{k}}^-$ in Eq.~\\eqref{energyspectrum_no_magnetic_field} is modified as\n\t\t\\begin{equation}\n\t\tE_{\\bm{k}}^- = \\epsilon (\\bm{k}) - \\sqrt{V_{\\rm AB}(\\bm{k})^2  + \\bm{g}(\\bm{k})^2+ \\bm{h}^2 \\pm 2 |\\lambda|  }, \\label{energyspectrum_with_magnetic_field}\n\t\t\\end{equation}\nwhere $\\lambda^2 = V_{\\rm AB}(\\bm{k})^2\\,\\bm{h}^2+ \\left[ \\bm{g} (\\bm{k})\\cdot \\bm{h}\\right]^2 $. The magnetic field not only lifts the Kramers degeneracy but also causes the nematicity through the coupling $\\left[ \\bm{g}_0 (\\bm{k}) \\cdot \\bm{h}\\right]^2$ in $\\lambda$, although linear terms in $\\bm{h}$ are canceled out between sublattices in sharp contrast to acentric systems studied before~\\cite{Ideue2017}. For \\bma{} with Dresselhaus-type staggered ASOC~\\cite{Manchon2019spin-orbit-torque_review}, the nematicity denoted by $\\varepsilon^{xy}$ is maximally induced by the magnetic field $\\bm{h}$ parallel to $[110]$ or $[1\\bar{1}0]$.\n\nWe expect nematicity-assisted dichroism in \\bma{} under the magnetic field $\\bm{h} \\parallel [110]$ from the above discussions. In numerically calculated NLC $\\sigma_\\text{D}^{z;zz}$ with rotating the magnetic field in the azimuthal plane, the dichroism with {\\it two-fold field-angle dependence} is clearly seen (Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}). In this case the magnetic field is a \\textit{bipolar field} rather than a vector field, in sharp contrast to the magnetic Drude term for which the observed field-angle dependence is one-fold~\\cite{Rikken2001magnetochiral_anisotropy,Rikken2005magnetoelectric_anisotropy,Ideue2017}. Although the field-induced NLC is tiny as evaluated in Appendix~\\ref{App_Sec_nematic_assisted_NLC}, it was actually detected in a recent experiment for \\bma{}~\\cite{KimataPrivate}.\n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{nonlinear_response_chemipot_-500_mev_allindex_hargEle_90_azimuthsweep_BaMnAs_unit_135.pdf}\n\t\t\\caption{Drude term of a longitudinal NLC $\\sigma^{z;zz}_\\text{D}$ as a function of the azimuthal angle of external magnetic fields $\\bm{h}=h(\\cos \\phi, \\sin \\phi, 0)$. Strength of the magnetic field $h=0.01$, temperature $T=0.01$, chemical potential $\\mu = -0.5$, relaxation time $\\gamma^{-1}=10^3$, and Brillouin zone mesh $N = 135^3$ are adopted. The other parameters and adopted energy scale are described in Appendix~\\ref{App_Sec_calc_NLC_Mn_magnet}.}\n\t\t\\label{Fig_drude_nematicity_assisted_azimuth_dependence} \n\t\t\\end{figure}\n\n\n\\subsection{Magnetic ASOC and Berry curvature dipole}\\label{Sec_magneticASOC_BCD}\n\nNow we consider the counterpart of the magnetic Drude term~\\cite{Ideue2017}, that is the magnetic BCD term. The \\PT{}-symmetry ensures Kramers doublet at each momentum $\\bm{k}$, and Berry curvature is completely canceled in the odd-parity magnetic multipole systems. The doublet, however, should be split when the \\PT{}-symmetry is broken by the external magnetic field. Let us consider \\bma{}-type magnet under the magnetic field $\\bm{h}=h_z \\hat{z}$ for an example. Then, while the total Berry curvature $\\int d\\bm{k} \\,\\Omega^z$ is trivially induced, the BCD also emerges. Using the allowed symmetry operations, the induced BCD is identified as\n\t\t\\begin{equation}\n\t\t\\mathcal{D}^{\\,xy} = \\mathcal{D}^{\\,yx}. \\label{induced_BCD_in_BMA_with_zField}\n\t\t\\end{equation}\n\nBecause the BCD has the same symmetry as the ASOC~\\cite{Manchon2019spin-orbit-torque_review}, emergence of one indicates the presence of the other.\nTherefore, the field-induced BCD is understood by discussing magnetically-induced ASOC in the following way. Although the sublattice-dependent ASOC is compensated with $\\bm{h}=0$, combination of the staggered exchange spitting $\\bm{h}_\\text{AF}\\cdot \\bm{\\sigma}~\\tau_z$ and uniform Zeeman field $\\bm{h} \\cdot \\bm{\\sigma}~\\tau_0$ leads to imbalance between the sublattices without Brillouin zone folding (Fig.~\\ref{Fig_magnetic_asoc}). One of sublattices obtains an increased carrier density, and consequently the sublattice-dependent ASOC is not compensated. The emergent ASOC has distinct properties compared to the conventional crystal ASOC since the former originates solely from the magnetic effects. We therefore name this field-induced ASOC `magnetic ASOC'. Interestingly, the magnetic ASOC is tunable by external magnetic fields. Thus, the concept of magnetic ASOC may be useful to design spin-momentum locking in more controllable way than the crystal ASOC which is determined by the crystal structure~\\cite{magASOC}. In the model for \\bma{} the magnetic ASOC and BCD with the same symmetry as Eq.~\\eqref{induced_BCD_in_BMA_with_zField} are actually obtained. \n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{magnetic_dresselhaus.pdf}\n\t\t\\caption{Mechanism of the magnetic ASOC and field-induced BCD. The blue-colored arrows denote the spin-polarization or Berry curvature at each $\\bm{k}$. (Left panel) A magnetic field along the $z$-axis splits the Fermi surface depending on the antiferromagnetic molecular field $\\bm{h}_\\text{AF}$. (Right panel) The split Fermi surface is viewed in the $xy$-plane which indicates the Dresselhaus-type ASOC and BCD.}\n\t\t\\label{Fig_magnetic_asoc} \n\t\t\\end{figure}\n\nThe field-induced BCD allows nonlinear Hall conductivity in accordance with Eq.~\\eqref{BCD_term}, which satisfies the relation \n\t\t\\begin{equation}\n\t\t\\sigma_\\text{BCD}^{z;xx} = -\\sigma_\\text{BCD}^{z;yy}= -2\\sigma_\\text{BCD}^{x;xz}= 2\\sigma_\\text{BCD}^{y;yz}.\n\t\t\\label{nonlinear_Hall_by_BCD}\n\t\t\\end{equation}\nFor example, we show the numerical result for $\\sigma_\\mathrm{BCD}^{z;xx}$ in Fig.~\\ref{Fig_BCD_with_z_field_elevation_dependence}, which reveals the dependence on the elevation angle of $\\bm{h}$. The induced BCD is inverted when the external field is flipped. Therefore, the field-angle dependence is one-fold in contrast to the nematicity-assisted dichroism (Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}).\n\n\t\t\\begin{figure}[htbp]\n\t\t\\centering \n\t\t\\includegraphics[width=75mm,clip]{nonlinear_response_chemipot_-500_mev_allindex_elevatesweep_Azu_0_BaMnAs_unit_135.pdf}\n\t\t\\caption{BCD term of a nonlinear Hall conductivity $\\sigma^{z;xx}_\\text{BCD}$ as a function of the elevation angle of external magnetic fields $\\bm{h}=h(\\sin \\theta, 0, \\cos \\theta)$. Parameters and unit are the same as Fig.~\\ref{Fig_drude_nematicity_assisted_azimuth_dependence}.}\n\t\t\\label{Fig_BCD_with_z_field_elevation_dependence} \n\t\t\\end{figure}\n\nFinally, we comment on a linear Hall response. Because the systems under the external magnetic field possess neither the \\T{}- nor \\PT{}-symmetry, a linear Hall response is also allowed. This is in contrast to the previously studied acentric systems~\\cite{Moore2010,Sodemann2015,Xu2018BCD_switchable,Ma2019BCD_experiment_WTe2} where the linear Hall response is forbidden because of the \\T{}-symmetry. However, the nonlinear Hall response can be distinguished from the linear one by symmetry. For example, the NLC, $\\sigma_\\text{BCD}^{z;xx}$ and $\\sigma_\\text{BCD}^{z;yy}$, in Eq.~\\eqref{nonlinear_Hall_by_BCD} represents the Hall response for which the linear response is forbidden.\n\n\\section{Conclusion and Discussions}\n\nThis work presents symmetry classification of the second-order NLC, and explores the NLC of odd-parity magnetic multipole systems. The Drude term gives rise to a giant nonlinear Hall conductivity at zero magnetic field, and provides an experimental tool for a probe of the sublattice-dependent ASOC. Thus, the hidden spin polarization in centrosymmetric crystals can be clarified. It also enables us to elucidate domain states in antiferromagnetic metals, and hence the NLC will be useful in the field of antiferromagnetic spintronics. Interestingly, the NLC induced by magnetic fields is significantly different from those studied in previous works. \nWe clarified the nematicity-assisted dichroism and the BCD-induced NLC due to the magnetic ASOC. \n\nIn accordance with our theoretical result, a recent experimental study actually detected nematicity-assisted electric dichroism under the magnetic field~\\cite{KimataPrivate}. \nWe believe that further studies of the nonlinear response in parity-violated magnetic systems will be motivated by our work.\n\n\n\\textbf{Acknowledgments}---\n\nThe authors are grateful to  A.~Shitade, A.~Daido,~Y. Michishita, M.~Kimata, and R.~Toshio for valuable comments and discussions. Especially, the authors thank M.~Kimata for providing experimental data and motivating this work. This work is supported by a Grant-in-Aid for Scientific Research on Innovative Areas ``J-Physics'' (Grant No.~JP15H05884) and ``Topological Materials Science'' (Grant No.~JP16H00991,~No,~JP18H04225) from the Japan Society for the Promotion of Science (JSPS), and by JSPS KAKENHI (Grant No.~JP15K05164, No.~JP15H05745, and No.~JP18H01178). H.W. is a JSPS research fellow and supported by JSPS KAKENHI (Grant No.~18J23115).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec::introduction}\n\nThis article develops a novel functional Bayesian network for modeling directed conditional independence and causal relationships of multivariate functional data, which arise in a wide range of applications. For example, learning brain effective connectivity networks from electroencephalogram (EEG) records is crucial for understanding brain activities and neuron responses. Another example is longitudinal medical studies where multiple clinical variables are recorded at possibly distinct time points across variables and\/or patients. Knowing causal dependence of these clinical variables may help physicians decide the right interventions. Functional data can also go beyond those defined on time domain e.g., spatial domain (environmental data, spatially-resolved genomics, etc).\n\nJoint analysis of multiple functional objects has attracted great attention in recent years with focuses mainly on reducing dimensionality and capturing functional dependence. For instance, \\cite{kowal2017bayesian} and \\cite{kowal2019integer} proposed to model time-ordered functional data through a time-varying parameterization for functional time series. Using basis transformation strategies, \\cite{zhang2016functional} built an autoregressive model for spatially correlated functional data, while \\cite{lee2018bayesian} modeled functional data in serial correlation semiparametrically. \\cite{chiou2014linear} developed a linear manifold model characterizing the functional dependence between multiple random processes.\n\n\\paragraph{Functional Graphical Models} In a similar but conceptually different manner, functional graphical models have been recently proposed to model conditional independence of multivariate functional data. Graphical models gives rise to compact probabilistic representation of high-dimensional data through the graph-encoded conditional independence constraints. One key challenge is that the graph is typically unknown and must be inferred from data. While graphical models have been extensively studied for vector- and matrix-variate data \\citep{yuan2007model, wang2009bayesian, leng2012sparse, ni2017sparse}, only recently have there been several developments for the functional data. \\cite{zhu2016bayesian} extended Markov and hyper Markov laws of decomposable undirected graphs for random vectors to those for random functions. \\cite{qiao2019functional} adopted the group lasso penalty on the precision matrix of coefficients extracted from the basis expansion of functions. \\cite{zapata2022partial} introduced the idea of partial separability to reduce the computational cost of \\cite{qiao2019functional}. \\cite{qiao2020doubly} further extended \\cite{qiao2019functional} and proposed to characterize the time-varying conditional independence of random functions through smoothing techniques. To relax the Gaussian process assumption of the aforementioned methods, \\cite{li2018nonparametric}, \\cite{solea2022copula}, and \\cite{lee2022nonparametric} proposed models based on additive conditional independence and copula Gaussian models.\n\nDespite these exciting developments of functional undirected graphical models, the work on functional \\textit{directed} graphical models is sparse. Generally, undirected graphs admit a different set of conditional independence constraints from directed graphs. For example, the directed graph in Figure \\ref{ex1} implies $X_2 \\perp X_3$ but $X_2 \\not\\perp X_3 | X_1$, yet there exists no undirected counterpart that admits the same set of conditional (in)dependence assertions. More importantly, causal discovery (i.e., generation of plausible causal hypotheses) is only possible with directed graphs given additional causal assumptions \\citep{pearl2000causality}. To the best of our knowledge, the functional structural equation model recently proposed by \\cite{lee2022functional} is the only work that infers directional relationships from multivariate functional data. However, as will become evident in Section \\ref{sec::fbn} and \\ref{sec::inference}, our model differs from theirs in several significant aspects.\n\n\\paragraph{Causal Discovery} As hinted earlier, one of the two important problems we intend to address in this work is discovering causality from functional observations. Causal discovery is one of the first steps to investigate the physical mechanism that governs the operation and dynamics of an unknown system. Given the learned causal knowledge, subsequent causal inference (e.g., deriving the interventional and counterfactual distributions) can be conducted under the celebrated do-calculus framework \\citep{pearl2000causality}. Therefore, inferring causal relationships potentially has more significant scientific impacts than learning associations since it may help answer fundamental questions about the nature. Bayesian networks paired with causal assumptions are among the most popular approaches in identifying unknown causal structure represented by a directed acyclic graph (DAG). One pressing obstacle of using Bayesian networks to discover causality from purely observational data is that in general, only Markov equivalence classes (MEC) can be learned based on conditional independence constraints alone. Causal interpretations of members in the same MEC can be drastically different, and, generally, only bounds on causal effects can be calculated \\citep{maathuis2009estimating}. For example, the three DAGs in Figure \\ref{ex2} constitute an MEC with the only conditional independence $X_2 \\perp X_3 | X_1$, but the causal directions are completely reversed in the last graph compared to the first one.\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{0.25\\textwidth}\n\\includegraphics[width=\\textwidth]{EX.pdf}\n\\caption{}\n\\label{ex1}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.72\\textwidth}\n\\includegraphics[width=\\textwidth]{MEC}\n\\caption{}\n\\label{ex2}\n\\end{subfigure}\n\\caption{Two Markov equivalence classes. (a) $X_2 \\perp X_3$. (b) $X_2 \\perp X_3 | X_1$.}\n\\end{figure}\n\nSince 2006, numerous researchers, however, have found that causal discovery (unique causal structure identification) is indeed possible with additional distributional assumptions on the data generating process, at least for finite-dimensional data. Examples include but are not limited to linear non-Gaussian models (LiNGAM, \\citealt{shimizu2006linear}), non-linear additive noise models \\citep{hoyer2008nonlinear}, and linear Gaussian models with equal error variances \\citep{peters2014identifiability}. See more related methods in a recent book of \\cite{peters2017elements}. Although remarkable progresses have been made in the causal discovery area for traditional finite-dimensional data, what remains lacking is method capable of discovering causality from general, purely observational, multivariate functional data. We remark that given a known causal graph, there are existing approaches that can be used to infer causal effects. For example, \\cite{lindquist2012functional} developed a causal mediation analysis framework where the treatment and outcome are scalars and the mediator is a univariate random function. Our scope is substantially different from this line of works in that we do not assume the causal graph to be known; in fact, learning the causal graph structure is precisely the focus of this paper.\n\n\\paragraph{Proposed Functional Bayesian Networks} We propose a novel functional Bayesian network model for multivariate functional data for which the conditional independence and causal relationships are represented by a DAG. As one would expect, the proposed functional Bayesian network factorizes over the DAG and respects all directed Markov properties (i.e., conditional independence constraints) encoded in the DAG via the notion of d-separation. Then for ease of exposition, we reformulate the proposed Bayesian network constructed in the functional space to an equivalent Bayesian network defined on the space of basis coefficients via basis expansion. Because in practice, functional data are almost always observed with noises, two essential ingredients are built in the proposed Bayesian networks to capture the functional dependence and to learn the causal structure. First, we capture the within-function dependence through a set of orthonormal basis functions chosen in a data-driven way. The resulting basis functions are interpretable and computationally efficient. Second, we encode the unknown causal structure by a structural equation model on the basis coefficients. Due to the equivalence of probability measures on the functional space and the space of basis coefficients, the conditional independence and causal relationships naturally transform back to the original random functions. To allow for unique DAG identification, we move away from the Gaussian process assumption often adopted by the existing functional graphical models and instead assume our random functions are generated from a discrete scale mixture of Gaussian distributions. We theoretically prove and empirically verify that the unique DAG identification is indeed possible even when the functions are observed with noises.\n\nTo conduct inference and uncertainty quantification from a finite amount of data, the proposed model is based on a Bayesian hierarchical formulation with carefully chosen prior distributions. Posterior inference is carried out through Markov chain Monte Carlo (MCMC). We perform simulation studies to demonstrate the capability of the proposed  model in recovering causal structure and key parameters of interest. A real data analysis with brain EEG records illustrates the applicability of the proposed framework in real world. We also apply the proposed model to a COVID-19 multivariate longitudinal dataset (shown in Section D of the Supplementary Material).\n\nThe rest of the paper is structured as follows. We provide an overview of Bayesian networks in Section \\ref{sec::overview}. The proposed functional Bayesian network is introduced in Section \\ref{sec::fbn}, which includes elaborations of the functional linear non-Gaussian model (Section \\ref{sec::FLiNG}) and the causal identifiability theory (Section \\ref{sec:ci}). Section \\ref{sec::inference} is devoted to Bayesian inference of the proposed model. We provide simulation studies and applications in Sections \\ref{sec::experiment} and \\ref{sec::eeg}, respectively. The main contributions of this paper are summarized in Section \\ref{sec::discussion} with some concluding remarks.\n \n\\section{Overview of Bayesian Networks} \\label{sec::overview}\n\nThroughout the paper, vectors and matrices are boldfaced whereas scalars and sets are not.\n\n\\paragraph{DAGs and Bayesian Networks} Let $\\bm{X} = (X_1, \\ldots, X_p)^T \\in \\mathcal{X}_1 \\times \\cdots \\times \\mathcal{X}_p$ denote a $p$-dimensional random vector. Denote $[m]: = \\{1, \\ldots, m\\}$ for any integer $m \\geq 1$. Let $\\bm{X}_S = (X_j)_{j\\in S}$ be a subvector of $\\bm{X}$ with $S\\subseteq [p]$. A DAG $G = (V, E)$ consists of a set of nodes $V = [p]$ and a set of directed edges represented by a binary adjacency matrix $\\bm{E} = (E_{j\\ell})$ where $E_{j\\ell} = 1$ if and only if $\\ell\\rightarrow j$ for $\\ell \\neq j\\in V$. DAGs do not allow directed cycles $j_0 \\to j_1 \\to \\cdots \\to j_k = j_0$. Each node $j \\in V$ represents a random variable $X_j \\in \\mathcal{X}_j$; we may use $j$ and $X_j$ interchangeably when no ambiguity arises. Each directed edge $\\ell \\to j$ and the lack thereof represent conditional dependence and independence of $X_\\ell$ and $X_j$, respectively. Note that although $X_j$ is often a scalar but it does not need to be. In fact, $X_j$ is a random function or an infinite dimensional random vector in this article. Denote $pa_G(j) = \\{\\ell\\in V: \\ell \\to j\\}$ the set of parents of $j$ in graph $G$. A Bayesian network (BN) $\\mathcal{B} = (G, P)$ on $\\bm{X}$ is a probability model where the joint probability distribution $P$ of $\\bm{X}$ factorizes with respect to $G$ in the following manner,\n\\begin{equation}\\label{eq:bnf}\nP(\\bm{X}) = \\prod_{j = 1}^p P_j(X_j | \\bm{X}_{pa_G(j)}),\n\\end{equation}\nwhere $P_j$ is the conditional distribution of $X_j$ given $\\bm{X}_{pa_G(j)}$ under $P$. Let $de_G(j) = \\{\\ell \\in V: j \\to \\cdots \\to \\ell\\}$ denote the descendants of $j$ in $G$ and let $nd_G(j) = V \\backslash de_G(j) \\backslash \\{j\\}$ denote the non-descendants of $j$. The BN factorization \\eqref{eq:bnf} directly implies the local directed Markov property -- any variable is conditionally independent of its non-descendants given its parents, $X_j \\perp \\bm{X}_{nd_G(j)\/pa_G(j)} | \\bm{X}_{pa_G(j)}, \\forall j \\in [p]$. In fact, the reverse is also true: if a distribution $P$ respects the local Markov property according to a DAG $G$, then $P$ must factorize over $G$ as in \\eqref{eq:bnf}. In summary, BN factorization and local Markov property are equivalent. We may omit the subscript $G$ of $pa_G(j)$ and $nd_G(j)$ and simply write $pa(j)$ and $nd(j)$ instead when $G$ is clear from the context.\n\n\\paragraph{Causal DAGs and Causal Bayesian Networks} A causal DAG $G$ is a DAG except that the directed edges are now interpreted causally, i.e., we say $X_\\ell$ is a direct cause (with respect to $V$) of $X_j$ and $X_j$ is a direct effect of $X_\\ell$ if $\\ell \\to j$. For simplicity, we will overload $nd(j)$ and $pa(j)$ to denote the noneffects and directed causes of $j$ in a causal DAG. To define a causal BN, we begin by asserting the local causal Markov assumption \\citep{spirtes2000causation,pearl2000causality} -- given a causal DAG $G$, a variable is conditionally independent of its noneffects given its direct causes. By noting the correspondence between noneffects and non-descendants, and between direct causes and parents in DAGs and causal DAGs, the local causal Markov assumption simply states that the distribution $P$ of $\\bm{X}$ respects the local Markov property of the causal DAG $G$, which in turn implies that $P$ must also factorize over $G$ (recall the equivalence between BN factorization and local Markov property). Therefore, a causal BN $\\mathcal{B} = (G, P)$ is a probability model where $P$ factorizes with respect to a causal DAG $G$ in the same way as in \\eqref{eq:bnf}.\n\n\\paragraph{Structural Equation Representation of Bayesian Networks} A BN is often represented by a structural equation model (SEM),\n\\begin{equation*}\nX_j = f_j(\\bm{X}, \\epsilon_j), ~ \\forall j \\in [p],\n\\end{equation*}\nwhere the transformation $f_j$ depends on $\\bm{X}$ only through its parents\/direct causes $\\bm{X}_{pa(j)}$, and the exogenous variables $\\bm{\\epsilon}=(\\epsilon_1,\\dots,\\epsilon_p)^T\\sim P_\\epsilon$ are assumed to be mutually independent. Denote the set of transformation functions as $F = \\{f_1, \\ldots, f_p\\}$. Since $F$ and $P_\\epsilon$ induce the joint distribution $P$ of $\\bm{X}$ and it is not difficult to show that the induced distribution $P$ factorizes over $G$, with a slight abuse of notation, we can rewrite the BN as $\\mathcal{B} = (G, F, P_\\epsilon)$.\n \n\\section{Functional Bayesian Networks} \\label{sec::fbn}\n\n\\subsection{General Framework} \\label{sec:gf}\n\nNow we introduce the construction of BNs for multivariate functional data. Denote the space of square integrable functions on domain $\\mathcal{D}$ with respect to measure $\\mu$ as $L^2(\\mathcal{D}) = \\{h: \\int_\\mathcal{D} h^2(\\omega) d\\mu(\\omega) < \\infty\\}$. We focus on a compact $\\mathcal{D} \\subset \\mathbb{R}$ (in fact, without loss of generality, $\\mathcal{D} = [0, 1]$) and the Lebesgue measure $\\mu$ for simplicity. Let $\\bm{Y} = (Y_1, \\ldots, Y_p)^T \\in L^2(\\mathcal{D}_1)\\times\\dots\\times L^2(\\mathcal{D}_p)$ be a collection of $p$ random functions. Denote $\\mathcal{H} = \\bigcup_{j = 1}^p \\{(\\omega, j): \\omega \\in \\mathcal{D}_j\\}$ the joint domain of $\\bm{Y}$ and $(L^2(\\mathcal{H}), \\mathcal{B}(L^2(\\mathcal{H})), P)$ its probability space. Similarly, for any subset $A \\subset [p]$, denote the joint domain $\\mathcal{H}_A = \\bigcup_{j \\in A} \\{(\\omega, j): \\omega \\in \\mathcal{D}_j\\}$ and $\\mathcal{B}(L^2(\\mathcal{H}_A))$ the Borel $\\sigma$-algebra on $L^2(\\mathcal{H}_A)$. Let $A, B, C$ be disjoint subsets of $[p]$. Following \\cite{zhu2016bayesian}, we say $\\bm{Y}_A$ is conditionally independent of $\\bm{Y}_B$ given $\\bm{Y}_C$ under $P$, if for any measurable set $D_A \\subset L^2(\\mathcal{H}_A)$, $P(\\bm{Y}_A \\in D_A | \\bm{Y}_B, \\bm{Y}_C)$ is $\\mathcal{B}(L^2(\\mathcal{H}_C))$ measurable and $P(\\bm{Y}_A \\in D_A | \\bm{Y}_B, \\bm{Y}_C) = P(\\bm{Y}_A \\in D_A | \\bm{Y}_C)$. We introduce a DAG $G = (V, E)$ where each node $j \\in V$ represents a random function $Y_j$. To begin with, we give the formal definition of a functional Bayesian network.\n\n\\begin{definition}[Functional Bayesian Networks]\nWe say $\\mathcal{B} = (G, P)$ is a functional Bayesian network for a set of random functions $\\bm{Y}$ if $P$ factorizes with respect to DAG $G$, \n\\begin{align*}\nP(Y_1 \\in D_1,\\dots,Y_p\\in D_p) = \\prod_{j = 1}^p P_j(Y_j \\in D_j | \\bm{Y}_{pa(j)} \\in D_{pa(j)}),\n\\end{align*}\nfor any measurable sets $D_j \\subset L^2(\\mathcal{D}_j), \\forall j \\in [p]$, where $P_j$ is the conditional probability measure of $Y_j$ given $\\bm{Y}_{pa(j)}$ under $P$.\n\\end{definition}\n\nJust like the ordinary finite-dimensional BN, the functional BN factorization implies the local Markov property and vice versa. \n\n\\begin{definition}[Functional Local Directed Markov Property]\nA probability measure $P$ of $\\bm{Y}$ satisfies the local directed Markov property with respect to $G$ if $Y_j \\perp \\bm{Y}_{nd(j)\/pa(j)} | \\bm{Y}_{pa(j)}$, i.e., $P(Y_j \\in D_j | \\bm{Y}_{nd(j)\/pa(j)}, \\bm{Y}_{pa(j)})$ is $\\mathcal{B}(L^2(\\mathcal{H}_{pa(j)}))$ measurable and $P(Y_j \\in D_j | \\bm{Y}_{nd(j)\/pa(j)}, \\bm{Y}_{pa(j)}) = P(Y_j \\in D_j | \\bm{Y}_{pa(j)})$ for any $D_j\\subset L^2(\\mathcal{D}_j)$.\n\\end{definition}\n\n\\begin{proposition}\nFunctional Bayesian network factorization is equivalent to functional local directed Markov property.\n\\end{proposition}\n\nProof is trivial. For modeling convenience, we use orthonormal basis expansion of random functions to (equivalently) redefine the functional BN in the space of basis coefficients. Let $\\{\\phi_{jk}\\}_{k = 1}^\\infty$ be a sequence of orthonormal basis functions of $L^2(\\mathcal{D}_j)$ and expand $Y_j = \\sum_{k = 1}^\\infty Z_{jk} \\phi_{jk}$, where $Z_{jk} = \\int_{\\mathcal{D}_j} Y_j(\\omega) \\phi_{jk}(\\omega) d\\omega$. The resulting coefficient sequence $\\bm{Z}_j = (Z_{jk})_{k = 1, \\ldots, \\infty}$ lies in the space of square summable sequences $\\ell_j^2 = \\{h_j: \\sum_{k = 1}^\\infty h_{jk}^2 < \\infty\\}$. The within-function and the between-function covariance can then be expressed in terms of the covariance of the coefficient sequences,\n\\begin{align*}\n\\text{cov}(Y_j(\\omega_j), Y_\\ell(\\omega_\\ell)) = \\sum_{k = 1}^\\infty \\sum_{h = 1}^\\infty \\phi_{jk}(\\omega_j) \\phi_{\\ell h}(\\omega_\\ell) \\text{cov}(Z_{jk}, Z_{\\ell h}), ~ \\forall \\omega_j \\in \\mathcal{D}_j, \\omega_\\ell \\in \\mathcal{D}_\\ell, ~ \\forall j, \\ell \\in [p].\n\\end{align*}\nBecause $L^2(\\mathcal{D}_j)$ and $\\ell_j^2$ are isometrically isomorphic for each $j$, for any disjoint subsets $A, B, C \\subset [p]$, $\\bm{Y}_A \\perp \\bm{Y}_B | \\bm{Y}_C$ if and only if $\\bm{Z}_A \\perp \\bm{Z}_B | \\bm{Z}_C$ where $\\bm{Z} = (\\bm{Z}_1, \\ldots, \\bm{Z}_p)^T$. Hence, if $\\bm{Y}$ follows the proposed BN model $\\mathcal{B} = (G, P)$, then the coefficient sequences $\\bm{Z}$ follows $\\mathcal{B}_Z = (G, P_Z)$ for some probability measure $P_Z$ of $\\bm{Z}$, and vice versa. Each node of the DAG $G$ either represents a random function $Y_j$ or, equivalently, its corresponding coefficient sequence $\\bm{Z}_j$. Moreover, the joint probability measure $P$ of $\\bm{Y}$ factorizes with respect to $G$ if and only if the joint probability measure $P_Z$ of $\\bm{Z}$ factorizes with respect to $G$. \n\n\\begin{proposition}\nSuppose $\\bm{Y}\\sim P$ and let $\\bm{Z}$ be the corresponding coefficient sequences from orthonormal basis expansion.\nThen \n\\begin{align*}\nP(Y_1 \\in D_1,\\dots,Y_p\\in D_p) = \\prod_{j = 1}^p P_j(Y_j \\in D_j | \\bm{Y}_{pa(j)} \\in D_{pa(j)}),\n\\end{align*}\nfor any measurable sets $D_j \\subset L^2(\\mathcal{D}_j), \\forall j \\in [p]$ if and only if\n\\begin{align*}\nP_Z(\\bm{Z}_1 \\in D_1',\\dots,\\bm{Z}_p\\in D_p') = \\prod_{j = 1}^p P_{Z j}(\\bm{Z}_j \\in D_j' | \\bm{Z}_{pa(j)} \\in D_{pa(j)}'),\n\\end{align*}\nfor any measurable sets $D_j' \\subset \\ell_j^2, \\forall j \\in [p]$.\n\\end{proposition}\n\nThe proof directly follows the preceding paragraph. Again, just like the ordinary finite-dimensional BN, if one makes the causal Markov assumption, the DAG $G$ in the proposed functional BN can be interpreted causally. Hereafter, by default, we always make the causal Markov assumption (hence $G$ is a causal DAG, the edge strength is interpreted as direct causal effect, etc) but all the results are simply reduced to those of a directed conditional independence model when the causal Markov assumption is dropped.\n \n\\subsection{Functional Linear Non-Gaussian Bayesian Networks} \\label{sec::FLiNG}\n\nSection \\ref{sec:gf} introduces a general framework for modeling directed conditional independence and causal relationships for multivariate functional data. In this subsection, we discuss in detail one specific case of the proposed general framework, namely the \\underline{F}unctional \\underline{Li}near \\underline{N}on-\\underline{G}aussian (FLiNG) BNs. Specifically, the FLiNG-BN assumes $\\bm{Z}$ follows a linear SEM,\n\\begin{align} \\label{eq1}\n\\bm{Z}_j = \\sum_{\\ell = 1}^p \\bm{B}_{j\\ell} \\bm{Z}_\\ell + \\bm{\\epsilon}_j, ~ \\forall j \\in [p],\n\\end{align}\nwhere $\\bm{\\epsilon}_j$ is an infinite-dimensional exogenous vector, $\\bm{B}_{j\\ell} = (B_{j\\ell}(k_j, k_\\ell))_{k_j=1,k_\\ell=1}^{\\infty,\\infty}$ is an infinite-dimensional direct causal effect matrix from $\\bm{Z}_\\ell$ to $\\bm{Z}_j$, and $\\ell\\to j$ is present in $G$ (i.e., $\\bm{Z}_\\ell$ is a direct cause of $\\bm{Z}_j$) if there exist $k_\\ell$ and $k_j$ such that $B_{j\\ell}(k_j, k_\\ell) \\neq 0$. Neither causal effects nor the causal graph is assumed to be known; therefore the main goal of this article is precisely to infer them from observational data. Because $L^2(\\mathcal{D}_j)$ and $\\ell_j^2$ are isometrically isomorphic for all $j \\in [p]$, the casual relationships of $\\bm{Z}$ encoded in DAG $G$ directly transfer to the the casual relationships of $\\bm{Y}$, i.e., $\\bm{Z}_\\ell$ is a direct cause of $\\bm{Z}_j$ if and if only if $Y_\\ell$ is a direct cause of $Y_j$.\n\nIn practice, the random functions $\\bm{Y}$ can only be measured on finite grids with random noises. In other words, we do not observe realizations of $\\bm{Y}$ but instead we observe realizations of $\\bm{W}=(\\bm{W}_1,\\dots,\\bm{W}_p)^T$ where $\\bm{W}_{j} = (W_{j}(1), \\ldots, W_{j}(m_j))$, which is the set of measurements of $Y_{j}$ on a finite grid $D_{j} = \\{\\omega_{j}(1), \\ldots, \\omega_{j}(m_j)\\} \\subset \\mathcal{D}_j$ with independent white noises $e_{j}(m) \\sim N(0, \\sigma_j)$, $\\forall m \\in [m_j]$, \n\\begin{align} \\label{eq2}\nW_j(m) = Y_{j}(\\omega_{j}(m)) + e_{j}(m).\n\\end{align}\nNote that $D_{j}$ can be different across $j$ (also across realizations).\n\nOne seemingly inconsequential element of the FLiNG-BN but turning out to be crucial for discovering causality is the specification of the probability distribution of the exogenous variables $\\bm{\\epsilon}_j= (\\epsilon_{jk})_{k=1}^\\infty$ in \\eqref{eq1}.\nA tempting choice may be Gaussian but it is the non-Gaussianity of $\\bm{\\epsilon}_j$'s that allows causal identification as we will show in Section \\ref{sec:ci}. Specifically, we assume $\\epsilon_{jk}$ to follow a finite scale mixture of Gaussian distributions, $\\epsilon_{jk} \\sim \\sum_{m = 1}^{M_{jk}} \\pi_{jkm} N(0, \\tau_{jkm})$,\nwhere $M_{jk}$ is the number of mixture components. The non-Gaussian exogenous variables lead to non-Gaussian coefficient sequences $\\bm{Z}$, which in turn lead to non-Gaussian-process distributed random functions $\\bm{Y}$. In addition to enabling causal identification, non-Gaussian-processes are robust against outlying curves \\citep{zhu2011robust}. For finite sample inference, we truncate the orthonormal basis at level $K$ such that $\\bm{\\phi}_j = (\\phi_{j1}, \\ldots, \\phi_{jK})^T$, as commonly done in existing functional data analysis literature. Consequently, \\eqref{eq2} is turned into\n\\begin{align} \\label{eq3}\nW_{j}(m) = \\sum_{k = 1}^K Z_{jk} \\phi_{jk}(\\omega_{j}(m)) + e_{j}(m).\n\\end{align}\n \n\\subsection{Causal Identifiability} \\label{sec:ci}\n\nThe proposed functional BNs are useful representations of directed conditional independence and causal relationships for multivariate functional data. The big remaining question is the learning of the underlying (causal) DAGs from observational data. Constraint-based methods, which are often model-free, have been popular for DAG learning. For the proposed functional BNs, we, in principle, can also use constraint-based methods, which test for conditional independence of pairs of functions. However, conditional independence tests are notoriously difficult and inefficient even for scalar random variables. Furthermore, even if we have access to oracle conditional independence tests for random functions, we can only hope for identifying the best MEC by definition (recall that an MEC contains DAGs with exactly the same set of conditional independence relationships). This may be acceptable if one is only interested in learning conditional independence relationships. But as mentioned in Section \\ref{sec::introduction}, for causal discovery, this is clearly unsatisfactory because the directionality of a potentially large number of edges of Markov equivalent DAGs may be left undetermined and hence the causal interpretations of these edges are unclear. Because the proposed FLiNG-BN is a proper probability model, we can exploit certain feature of the model, namely the non-Gaussianity, to uniquely identify the underlying causal DAG.\n \n\\begin{definition}[Causal Identifiability]\nSuppose $\\bm{Y}$ follows the FLiNG-BN $\\mathcal{B} = (G, P)$, and suppose $\\bm{W}$ is a noisy version of $\\bm{Y}$ with noise variances $\\bm{\\sigma}= (\\sigma_1, \\ldots, \\sigma_p)$ as defined in \\eqref{eq2}. Let $P_W$ denote the distribution of $\\bm{W}$ induced from FLiNG-BN and the noises. We say that the causal DAG of FLiNG-BN is identifiable from $\\bm{W}$ if there does not exist another BN $\\mathcal{B}' = (G', P')$ where $G'\\neq G$ and noise variances $\\bm{\\sigma}' = (\\sigma'_1, \\ldots, \\sigma'_p)$ such that the induced distributions on $\\bm{W}$, $P'_W$, is equivalent to $P_W$, i.e., $P_W(\\bm{W}) \\equiv P'_W(\\bm{W})$.\n\\end{definition}\n\n\\begin{theorem}[Causal Identifiability] \\label{them1}\nThe causal DAG of FLiNG-BN is identifiable if the number of Gaussian mixture components $M_{jk} > 1, \\forall j, k$.\n\\end{theorem}\n\nTheorem \\ref{them1} signifies that by examining the probability distribution $P_W$, to which we have access through the observational data alone, one can gauge the likelihood that a given causal DAG is the data generating DAG. With a finite dataset, we shall focus on weighing different candidate causal DAGs by their posterior probabilities. Here, we provide the outline of the proof; the complete proof is given in Section A of the Supplementary Material. Given a chosen set of basis functions, we show the result in the space of basis coefficients. The problem then transforms to prove that, given $\\bm{Z} = \\bm{B} \\bm{Z} + \\bm{\\epsilon}$ and observe $\\bm{W} = \\bm{Z} + \\bm{e}$, there does not exist another equivalent parameterization $\\bm{Z}' = \\bm{B}' \\bm{Z}' + \\bm{\\epsilon}'$ and $\\bm{W} = \\bm{Z}' + \\bm{e}'$. Since we assume each component of $\\bm{\\epsilon}$ follows a Gaussian scale mixture, the induced distribution on $\\bm{W}$ is a multivariate Gaussian mixture (with different precision matrices). We then prove the causal effect matrix $\\bm{B}$ is uniquely identifiable from such mixture model by combining the identification of Gaussian mixture components, uniqueness of LDL decomposition, and proof of causal ordering identification. We demonstrate the identifiability result with a toy example.\n\n\\begin{example} \\label{exp1}\nConsider a true functional causal graph $1\\to 2$ and the corresponding data generating model $Z_1 = \\epsilon_1$ with $\\epsilon_1 \\sim 0.5 N(0, 0.5) + 0.5 N(0, 1)$ and $Z_2 = Z_1 + \\epsilon_2$ with $\\epsilon_2 \\sim 0.5 N(0, 0.5) + 0.5 N(0, 1)$; note for simplicity, we assume in this example that the number of basis functions is $K=1$. Assume we observe with noises $W_1 = Z_1 + e_1$ and $W_2 = Z_2 + e_2$ with $e_1, e_2 \\sim N(0, 0.1)$. We sample $n = 1000$ observations from this model and index them by the subscript $i=1,\\dots,n$. For the purpose of illustration, suppose we know the mixture component assignment of each observation and define four groups of observations based on the combination of variances of $\\epsilon_1$ and $\\epsilon_2$,\n\\begin{equation*}\n\\begin{aligned}\n& C_1 = \\{i: \\text{Var}(\\epsilon_{i1}) = 0.5 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 0.5\\}, ~ C_2 = \\{i: \\text{Var}(\\epsilon_{i1}) = 0.5 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 1\\}, \\\\\n& C_3 = \\{i: \\text{Var}(\\epsilon_{i1}) = 1 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 0.5\\}, ~ C_4 = \\{i: \\text{Var}(\\epsilon_{i1}) = 1 ~\\text{and}~ \\text{Var}(\\epsilon_{i2}) = 1\\}.\n\\end{aligned}\n\\end{equation*}\nWe fit linear regression separately to all observations and to observations in each of the four groups with the true causal direction $1 \\to 2$ (regressing $W_2$ on $W_1$) and the  anti-causal direction $2 \\to 1$ (regressing $W_1$ on $W_2$), which are shown in Figure \\ref{demo}. We observe that the fitted lines are almost identical in the causal direction for all groups whereas they can be quite different across groups in the anti-causal direction. Therefore, only the true causal graph gives a unique regression coefficient among all groups. Notice that if there is only one mixture component (i.e., degeneration to the Gaussian case), no comparison can be made between the causal and anti-causal directions since there will be only one regression line.\n\\end{example}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1\\linewidth]{DEMO}\n\\caption{A toy example for demonstration of causal identification. The left (right) panel shows the linear regression of $W_2$ ($W_1$) on $W_1$ ($W_2$). Data are simulated from the causal graph $1\\to 2$. Colored lines are the fitted linear regressions for all observations and for the observations in groups $C_1$--$C_4$.}\n\\label{demo}\n\\end{figure}\n\nThe next counter example illustrates the necessity of the non-Gaussian assumption for causal identification.\n\n\\begin{example}\nConsider a similar bivariate case to Example \\ref{exp1} but now the exogenous variables are Gaussian instead of mixture of Gaussian. Suppose the true functional causal graph $1\\to 2$ and the corresponding data generating model $Z_1 = \\epsilon_1$ with $\\epsilon_1 \\sim N(0, \\tau_1)$ and $Z_2 = b Z_1 + \\epsilon_2$ with $\\epsilon_2 \\sim N(0, \\tau_2)$; note again for simplicity, we assume in this example that the number of basis functions is $K = 1$. Assume we observe with noises $W_1 = Z_1 + e_1$ and $W_2 = Z_2 + e_2$ with $e_1 \\sim N(0, \\sigma_1)$ and $e_2 \\sim N(0, \\sigma_2)$. The induced joint distribution on $\\bm{W} = (W_1, W_2)$ is then bivariate Gaussian with mean $0$ and covariance matrix\n$$\\begin{pmatrix}\n\\tau_1 + \\sigma_1 & b \\tau_1\\\\\nb \\tau_1 & b^2 \\tau_1 + \\tau_2 + \\sigma_2\n\\end{pmatrix}. $$\n\nFurther consider the anti-causal model $2 \\to 1$ where $Z_2' = \\epsilon_2'$ with $\\epsilon_2' \\sim N(0, \\tau_2')$ and $Z_1' = b' Z_2' + \\epsilon_1'$ with $\\epsilon_1' \\sim N(0, \\tau_1')$. Suppose $W_1 = Z_1' + e_1'$ and $W_2 = Z_2' + e_2'$ with $e_1' \\sim N(0, \\sigma_1')$ and $e_2' \\sim N(0, \\sigma_2')$. The induced joint distribution on $\\bm{W} = (W_1, W_2)$ is still bivariate Gaussian with mean 0 and covariance matrix\n$$\\begin{pmatrix}\nb^{'2} \\tau_2' + \\tau_1' + \\sigma_1' & b' \\tau_2'\\\\\nb' \\tau_2' & \\tau_2' + \\sigma_2'\n\\end{pmatrix}. $$ \nFor any chosen $\\tau_1', \\sigma_1' > 0$ such that $\\tau_1' + \\sigma_1' < \\tau_1 + \\sigma_1 - b^2 \\tau_1^2 \/ (b^2 \\tau_1 + \\tau_2 + \\sigma_2)$, if we set\n\\begin{equation*}\n\\begin{aligned}\n& b' = (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1') \/ b \\tau_1, \\\\\n&\\tau_2' = b^2 \\tau_1^2 \/ (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1'), \\\\\n& \\sigma_2' = b^2 \\tau_1 + \\tau_2 + \\sigma_2 - b^2 \\tau_1^2 \/ (\\tau_1 + \\sigma_1 - \\tau_1' - \\sigma_1'),\n\\end{aligned}\n\\end{equation*}\nthen the induced distribution coincides with that under the true causal model (i.e., Gaussian with mean 0 and the same covariance). Therefore, causal identification fails in this case.\n\\end{example}\n\n\\section{Bayesian Inference} \\label{sec::inference}\n\nThe inference of the proposed FLiNG-BN framework can be carried out in either a frequentist (e.g., maximizing penalized likelihood) or a Bayesian (e.g., sampling from posterior distribution) fashion. Existing frequentist functional graphical models \\citep{qiao2019functional,qiao2020doubly,zapata2022partial,solea2022copula,lee2022nonparametric,lee2022functional} often estimate graphs in two separate steps -- (i) estimate the basis coefficient sequence of each function marginally via functional principle component analysis, and (ii) learn an directed\/undirected graph based on the estimated coefficient sequences. However, the eigenfunctions that marginally explain the most variation of each individual function do not necessarily explain well the conditional\/causal relationships among a set of functions. Moreover, the estimation uncertainty is not propagated from the first step to the second, which may result in overly confident inference. To mitigate these potential drawbacks of the two-step approaches, we propose a fully Bayesian inference procedure that jointly infers basis coefficient sequences and the DAG structure. This joint inference approach constructs orthonormal basis functions adaptive to their conditional\/causal relationships and allows for finite-sample inference and uncertainty quantification. \n\n\\subsection{Adaptive Orthonormal Basis Functions}\n\nWe assume the basis functions to be shared across all random functions \\citep{kowal2017bayesian,zapata2022partial}, $\\phi_{jk}(\\omega) := \\phi_k(\\omega),\\forall j\\in [p]$, which are more parsimonious than models based on function-specific basis functions. Moreover, the common basis functions put the basis coefficient sequences $\\bm{Z}_j,\\forall j\\in[p]$ on an equal footing (e.g., the magnitudes of basis coefficients are directly comparable) so that they are directly comparable and the BN on $\\bm{Z}$ has a more coherent interpretation. In this case, the non-zero matrix block $\\bm{B}_{j\\ell}$ corresponds to a causal connection from $\\bm{Y}_\\ell$ to $\\bm{Y}_j$. Loosely speaking, if we regard the basis functions as signal channels, then a significant non-zero $B_{j\\ell}(k_j, k_\\ell)$ indicates that $\\bm{Y}_\\ell$ directly affects $\\bm{Y}_j$ through its signal transmission from the $k_\\ell$-th channel to the $k_j$-th channel. \n\nAs mentioned above, we do not pre-specify a fixed set of orthonormal basis functions but instead they are learned adaptively from data by further expanding them with spline basis functions \\citep{kowal2017bayesian}, $\\phi_k(\\omega) = \\sum_{\\ell = 1}^L A_{k\\ell} b_\\ell(\\omega)$, where $\\bm{b} = (b_1, \\ldots, b_L)^T$ is a set of cubic B-spline basis functions with equally spaced knots and $\\bm{A}_k = (A_{k1}, \\ldots, A_{kL})^T, \\forall k \\in [K]$ are spline coefficients. Because $\\bm{A}_k$'s are not fixed \\emph{a priori}, so are $\\phi_k$'s.\n\n\\subsection{Prior Model}\n\nWe summarize our model and its entailed parameters using a DAG shown in Figure \\ref{hmodel}. The prior distributions of the model parameters are introduced in this section. We simulate posterior samples through Markov chain Monte Carlo (MCMC). Details are given in Section B of the Supplementary Material. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1\\linewidth]{HMODEL}\n\\caption{A DAG illustrating the model hierarchy. Single-line arrows are stochastic relationships and double-line arrows are deterministic relationships. The observed node $\\bm{W}_j$ is shown in rectangle and other nodes are shown in circles.}\n\\label{hmodel}\n\\end{figure}\n\n\\paragraph{Prior on B-spline Coefficients $\\bm{A}_k$} The prior on $\\bm{A}_k$ serves three purposes. First, it forces $\\phi_k$'s to be orthonormal, i.e., $\\int \\phi_k(\\omega) \\phi_h(\\omega) d\\omega = I(k = h), \\forall k, h \\in[K]$. Second, it regularizes the roughness of $\\phi_k$'s to prevent overfitting and sorts the orthonormal basis functions by increasing roughness. Third, it enables posterior inference on the orthonormal basis functions simultaneously with the graph estimation without having to fix them \\emph{a priori}.\n\nWe summarize the main steps of prior specification and refer the details to \\cite{kowal2017bayesian}. First, to regularize the roughness of $\\phi_k$ in a frequentist framework, one would consider a penalized likelihood with the roughness penalty,\n$$\\lambda_k \\mathcal{P}(\\bm{A}_k) = \\lambda_k \\int [\\phi_k^{''}(\\omega)]^2 \\ d\\omega = \\lambda_k \\bm{A}_k^T \\bm{\\Omega} \\bm{A}_k,$$\nwhere $\\lambda_k > 0$ is the regularization parameter and $\\bm{\\Omega} = \\int \\bm{b}^{''}(\\omega)[\\bm{b}^{''}(\\omega)]^T \\ d\\omega$. As a Bayesian counterpart, the regularization term is equivalent to a prior on the B-spline coefficients $\\bm{A}_k \\sim N(\\bm{0}, \\lambda_k^{-1} \\bm{\\Omega}^{-})$, where $\\bm{\\Omega}^{-}$ is a pseudo-inverse (since $\\bm{\\Omega}$ is rank-deficient by 2). Let $\\bm{\\Omega} = \\bm{U} \\bm{D} \\bm{U}^T$ be the singular value decomposition of $\\bm{\\Omega}$. To facilitate efficient computation, we follow \\cite{wand2008semiparametric} and reparameterize $\\phi_k = \\sum_{\\ell=1}^L A_{k\\ell}b_\\ell = \\sum_{\\ell=1}^L \\tilde{A}_{k\\ell} \\tilde{b}_\\ell$ with $\\tilde{\\bm{b}}(\\omega) = (1, \\omega, \\bm{b}^T(\\omega) \\bm{U}_P \\bm{D}_P^{-1\/2})^T$ where $\\bm{D}_P$ is the $(L - 2) \\times (L - 2)$ submatrix of $\\bm{D}$ corresponding to non-zero singular values and $\\bm{U}_P$ is the corresponding $L \\times (L - 2)$ submatrix of $\\bm{U}$. The reparameterization induces a prior on $\\tilde{\\bm{A}}_k = (\\tilde{A}_{k1}, \\ldots, \\tilde{A}_{kL})^T\\sim N(0, S_k)$ where $S_k = \\mathrm{diag}(\\infty, \\infty, \\lambda_k^{-1}, \\ldots, \\lambda_k^{-1})$ with the first two dimensions corresponding to the unpenalized constant and linear terms. In practice, one can replace $\\infty$ by a large number, say $10^8$. \n\nSecond, we constrain the regularization parameters $\\lambda_1 > \\cdots > \\lambda_K > 0$ to identify the ordering of basis functions, which sorts the basis functions by decreasing smoothness. Unlike the functional principal component analysis (PCA) where the principal components are ordered by the proportion of variance explained, the adopted Bayesian approach is less prone to rough functions. Given the ordering constraint, a uniform prior is imposed such that $\\lambda_k \\sim U(L_k, U_k)$, where $U_1 = 10^8$, $L_k = \\lambda_{k + 1}$ for $k = 1, \\ldots, K - 1$, $U_k = \\lambda_{k - 1}$ for $k = 2, \\ldots, K$, and $L_K = 10^{-8}$.\n\nFinally, consider the orthonormal constraint\n\\begin{align} \\label{penmt}\n\\int \\phi_k(\\omega) \\phi_h(\\omega) = \\int \\tilde{\\bm{A}}_k^T \\tilde{\\bm{b}}(\\omega) \\tilde{\\bm{b}}^T(\\omega)\\tilde{\\bm{A}}_h \\ d\\omega = \\tilde{\\bm{A}}_k^T \\bm{J} \\tilde{\\bm{A}}_h = I(k = h),\n\\end{align}\nwith $\\bm{J} = \\int \\tilde{\\bm{b}}(\\omega) \\tilde{\\bm{b}}^T(\\omega) \\ d\\omega$. This constraint can be easily enforced by projection and normalization during the course of MCMC; see Section B of the Supplementary Material for details.\n\n\\paragraph{Priors on the DAG Adjacency Matrix $\\bm{E}$ and Direct Causal Effects $\\bm{B}$} The key problem we aim to address in this article is causal structure learning, i.e., inferring the adjacency matrix, $\\bm{E} = (E_{j\\ell})$ (recall $E_{j \\ell} = 1$ if and only if $\\ell \\to j$). We propose to use a beta-Bernoulli-like prior $E_{j \\ell} \\sim \\mathrm{Bernoulli}(r)$ with $r \\sim \\mathrm{Beta}(a_r, b_r)$, subject to the acyclicity constraint,\n$$P(\\bm{E} | r) \\propto \\prod_{j\\neq\\ell} r^{E_{j\\ell}} (1-r)^{1-E_{j\\ell}} I(G \\text{ is a DAG}).$$\nWe set $a_r = b_r = 1$. \\cite{scott2010bayes} showed that the beta-Bernoulli prior allows automatic multiplicity adjustment in sparse regression problem. In our context, the marginal distribution of $\\bm{E}$ with $r$ integrated out equals\n\\begin{align} \\label{mar}\nP(\\bm{E}) \\propto \\mathrm{Beta}\\left(\\sum_{j\\neq\\ell}E_{j\\ell}+1,\\sum_{j\\neq\\ell}(1-E_{j\\ell})+1\\right)I(G \\text{ is a DAG}).\n\\end{align}\nThe marginal distribution strongly prevents false discoveries by increasing the penalty against additional edges as the dimension $p$ grows. For example, the marginal \\eqref{mar} favors an empty graph over a graph with one edge by a factor of $p^2 - p$, which increases with $p$.\n\nConditional on $\\bm{E}$, we assume independent matrix-variate spike-and-slab priors on the direct causal effects,\n\\begin{align*}\n\\bm{B}_{j \\ell}|E_{j \\ell} \\sim (1 - E_{j \\ell}) \\delta_{\\bm{O}}(\\bm{B}_{j \\ell}) + E_{j \\ell} N(\\bm{B}_{j\\ell}|\\bm{O}, \\gamma \\bm{I}, \\bm{I}),\n\\end{align*}\nwhere $\\delta_{\\bm{O}}(\\cdot)$ is a point mass at a $K\\times K$ zero matrix $\\bm{O}$ and $N(\\cdot|\\bm{O}, \\gamma \\bm{I}, \\bm{I})$ is a centered matrix-variate normal distribution with row and column covariance matrices $\\gamma \\bm{I}$ and $\\bm{I}$ where $\\bm{I}$ is a $K\\times K$ identity matrix. The hyperparameter $\\gamma$ indicates the overall causal effect size and is assumed to follow a conjugate inverse-gamma prior, $\\gamma \\sim IG(a_\\gamma, b_\\gamma)$ with $a_\\gamma = b_\\gamma = 1$.\n\n\\paragraph{Prior on the Gaussian Scale Mixture} We choose conjugate priors,\n\\begin{align*}\n\\bm{\\pi}_{jk} = (\\pi_{jk1}, \\ldots, \\pi_{jkM}) \\sim \\text{Dirichlet}(\\alpha,\\dots,\\alpha), ~ \\tau_{jkm} \\sim IG(a_\\tau, b_\\tau), ~ \\forall j \\in [p], k \\in [K], m \\in [M],\n\\end{align*}\nwhich allows for straightforward Gibbs sampling. As default, we set $\\alpha = 1$ and $a_\\tau = b_\\tau = 1$. \n\n\\paragraph{Prior on Observation Noises} We complete the prior specification with a conjugate inverse-gamma prior on the variance of observation noises, $\\sigma_j \\sim IG(a_\\sigma, b_\\sigma), \\forall j \\in [p]$ with $a_\\sigma = b_\\sigma = 0.01$.\n\nFinally, we summarize the differences between the proposed FLiNG-BN and the work from \\cite{lee2022functional}. First, \\cite{lee2022functional} assume  their functions to be noiseless whereas we consider the scenario where functions are observed with noises. The causal identifiability theory is significantly more complicated when functions are noisy. Second, they assume their functions to be Gaussian whereas our functions are non-Gaussian; this difference leads to different learning algorithms and identifiability theory. Third, their inference is a two-step procedure based on causal ordering identification and sparse function-on-function regression, while the proposed Bayesian hierarchical model admits one-step inference procedure, which learns the graph structure by directly searching in the graph space without having to learn the causal ordering first.\n\n\\section{Simulation Studies} \\label{sec::experiment}\n\nWe conducted simulation studies to evaluate the proposed FLiNG-BN model. We considered two scenarios. In the first scenario, the functions were observed on an evenly spaced grid; this is the scenario commonly studied in the existing functional undirected graphical models \\citep{qiao2019functional} and is also similar to our later EEG application. In the second scenario, the functions were observed on an unevenly spaced grid, similar to the COVID-19 longitudinal application (the details are shown in Section C of the Supplementary Material). We compared the proposed FLiNG-BN with a functional undirected graphical model (FGLASSO; \\citealt{qiao2019functional}). We did not make comparison with \\cite{lee2022functional} due to lack of publicly available code at the time of submission. In addition, we compared FLiNG-BN with approaches based on two-step estimation procedures. In the first step, we extracted basis coefficients obtained from functional PCA using the package \\texttt{fdapace} \\citep{carroll2021}. In the second step, given the estimated basis coefficients, we constructed causal graphs using either the LiNGAM \\citep{shimizu2006linear} algorithm (termed FPCA-LiNGAM) or the PC \\citep{spirtes1991algorithm} algorithm (termed FPCA-PC). LiNGAM estimates a causal DAG based on the linear non-Gaussian assumption whereas PC generally returns only an equivalence class of DAGs based on conditional independence tests. Their implementations are available from R package \\texttt{pcalg} \\citep{kalisch2020overview}.\n\nTo mimic the EEG data application, we simulated data from FLiNG-BN with all the combinations of sample size $n \\in \\{50, 100, 200\\}$, number of functions $p \\in \\{30, 60, 90\\}$, and grid size $d \\in \\{125, 250\\}$. The grid spanned the unit interval $[0,1]$. We set the true number of basis functions to be $K = 5$. To generate basis functions, we first simulated the non-orthnormal functions $\\phi_k^U, \\forall k \\in [K]$ from a set of $L = 6$ cubic B-spline basis functions with evenly spaced knots, $\\phi_k^U = \\sum_{\\ell=1}^L A_{k\\ell}b_\\ell$, where $A_{k\\ell}$'s were generated from a standard normal distribution. We then empirically orthonormalized $(\\phi_1^U, \\ldots, \\phi_K^U)$ to get the orthonormal basis functions $(\\phi_1, \\ldots, \\phi_K)$. The simulation true causal graph $G$ was generated from the Erd\\H{o}s-R\\'{e}nyi model with connection probability $2\/p$, subject to the acyclicity constraint. Given the true graph $G$, each block of non-zero direct causal effects $\\bm{B}_{j\\ell}$ was generated independently from a standard matrix-variate normal distribution. Then the basis coefficient sequences $\\bm{Z}$ were generated from \\eqref{eq1} where the exogenous variables $\\bm{\\epsilon}_j$'s were generated from a centered Laplace distribution with scale $b = 0.5$. Note that when we fit FLiNG-BN to the simulated data, we still assumed the exogenous variables to be discrete scale Gaussian mixture although the simulation true exogenous variables were Laplace (i.e., continuous scale Gaussian mixture). Finally, noisy observations were simulated following \\eqref{eq3} with the signal-to-noise ratio, i.e., the mean value of $|y_j^{(i)}(\\omega_j^{(i)}(m))| \/ \\sigma_j$ across all samples $i \\in [n]$ and grid points $m \\in [m_{j}^{(i)}]$, set to 5.\n\nFor implementing the proposed FLiNG-BN, we set the number of mixture components to $M = 5$ and the number B-spline basis functions to $L=20$ (note that the simulation truth was $L=6$), and ran MCMC for 5,000 iterations (discarding the first half as burn-in and retaining every 5th iteration after burn-in). The causal graph $G$ was estimated by thresholding the posterior probability of inclusion at 0.5 (i.e., the median probability model). Parameters of competing methods were set to their default values. To assess the graph recovery performance, we calculated true positive rate (TPR), false discovery rate (FDR), and Matthews correlation coefficient (MCC),\n\\begin{align*}\n& \\text{TPR} = \\text{TP}(\\text{TP} + \\text{FN})^{-1}, ~~~~~ \n\\text{FDR} = \\text{FP}(\\text{TP} + \\text{FP})^{-1}, \\\\\n& \\text{MCC} = (\\text{TP} \\times \\text{TN} - \\text{FP} \\times \\text{FN})\\left[(\\text{TP} + \\text{FP}) \\times (\\text{TP} + \\text{FN}) \\times (\\text{TN} + \\text{FP}) \\times (\\text{TN} + \\text{FN})\\right]^{-1\/2},\n\\end{align*}\nwhere TP, TN, FP, and FN stand for the numbers of true positives, true negatives, false positives, and false negatives, respectively. MCC ranges from $-$1 to 1 with $0$ indicating a random guess and $1$ a perfect recovery. Since FGLASSO learns an undirected graph, we compared it with a moralization of the true graph\\footnote{Graph moralization converts a DAG to an undirected graph by first marrying all the unmarried parents and then removing all the directions. A probability distribution that respects the Markov property of a DAG must respect the Markov property of its moral graph.}. Similarly, since PC algorithm returns the MEC representation\\footnote{The MEC representation is shown in an essential graph, where any edge presented between two nodes is directed if and only if it follows the same direction in all members of this MEC. Otherwise, it is undirected.}, we compared it with the MEC of the true causal graph. \n\nThe results based on 50 repeat simulations are summarized in Table \\ref{s1}, from which we conclude that the proposed FLiNG-BN significantly outperformed all the competitors FGLASSO, FPCA-LiNGAM, and FPCA-PC across all combinations of $n$, $p$, and $d$. This is not surprising because (i) FGLASSO is not designed for learning directed graphs; they were compared with the proposed FLiNG-BN because of lack of alternative functional BN implementation. (ii) Although FPCA-LiNGAM and FPCA-PC are capable of learning directed graphs, they still performed poorly because they are implemented in a two-step procedure where there is little reason to believe that the basis coefficients extracted by the functional PCA in the first step are useful to capture the functional dependence in the second step. (iii) Unlike the proposed approach, none of the competing methods controls for false discovery and some impose the stringent Gaussian assumption, resulting in high FDR and\/or low TPR. Our suggested method to determine $K$ also worked well.\n\n\\begin{table}[ht]\n\\caption{Functions observed on evenly spaced grid. Average operating characteristics based on 50 repetitions are reported; standard deviations are given within the parentheses. Since LiNGAM is not applicable to cases where $q > n$ with $q = pK$ being the total number of extracted basis coefficients across all functions, the results from those cases are not available and indicated by -.}\n\\resizebox{\\textwidth}{!}{\n\\centering\n\\begin{tabular}{ccc|ccc|ccc|ccc|ccc}\n\\toprule\n\\multirow{2}{*}{$p$} & \\multirow{2}{*}{$d$} & \\multirow{2}{*}{$n$} & \\multicolumn{3}{c|}{FLiNG-BN} & \\multicolumn{3}{c|}{FGLASSO} & \\multicolumn{3}{c|}{FPCA-LiNGAM} & \\multicolumn{3}{c}{FPCA-PC} \\\\\n\\cmidrule(lr){4-6} \\cmidrule(lr){7-9} \\cmidrule(lr){10-12} \\cmidrule(lr){13-15}\n& & & TPR & FDR & MCC & TPR & FDR & MCC & TPR & FDR & MCC & TPR & FDR & MCC \\\\\n\\midrule\n30 & 125 & 50 & 0.62 (0.07) & 0.14 (0.07) & 0.72 (0.07) & 0.58 (0.02) & 0.88 (0.02) & 0.16 (0.02) & - & - & - & 0.22 (0.03) & 0.89 (0.02) & 0.12 (0.02) \\\\\n30 & 125 & 100 & 0.71 (0.08) & 0.19 (0.05) & 0.75 (0.06) & 0.63 (0.03) & 0.85 (0.03) & 0.20 (0.03) & 0.84 (0.02) & 0.85 (0.01) & 0.31 (0.02) & 0.30 (0.01) & 0.89 (0.01) & 0.13 (0.01) \\\\\n30 & 125 & 200 & 0.73 (0.05) & 0.13 (0.08) & 0.79 (0.06) & 0.69 (0.03) & 0.84 (0.05) & 0.19 (0.03) & 0.92 (0.04) & 0.87 (0.01) & 0.30 (0.01) & 0.13 (0.02) & 0.96 (0.01) & 0.02 (0.01) \\\\\n\\midrule\n30 & 250 & 50 & 0.68 (0.05) & 0.25 (0.08) & 0.73 (0.06) & 0.57 (0.02) & 0.88 (0.04) & 0.16 (0.04) & - & - & - & 0.30 (0.02) & 0.87 (0.01) & 0.15 (0.01) \\\\\n30 & 250 & 100 & 0.75 (0.04) & 0.26 (0.03) & 0.74 (0.03) & 0.64 (0.03) & 0.85 (0.04) & 0.18 (0.03) & 0.88 (0.04) & 0.86 (0.02) & 0.34 (0.02) & 0.18 (0.02) &0.92 (0.02) & 0.08 (0.01) \\\\\n30 & 250 & 200 & 0.85 (0.01) & 0.30 (0.06) & 0.79 (0.04) & 0.69 (0.02) & 0.83 (0.04) & 0.21 (0.02) & 0.97 (0.05) & 0.85 (0.03) & 0.35 (0.03) & 0.22 (0.02) & 0.94 (0.01) & 0.08 (0.02) \\\\\n\\midrule\n60 & 125 & 50 & 0.68 (0.03) & 0.05 (0.03) & 0.80 (0.02) & 0.57 (0.04) & 0.89 (0.06) & 0.11 (0.05) & - & - & - & 0.28 (0.02) & 0.87 (0.01) & 0.16 (0.01) \\\\\n60 & 125 & 100 & 0.68 (0.04) & 0.12 (0.04) & 0.75 (0.04) & 0.60 (0.03) & 0.85 (0.05) & 0.15 (0.04) & - & - & - & 0.28 (0.01) & 0.89 (0.01) & 0.15 (0.01) \\\\\n60 & 125 & 200 & 0.74 (0.02) & 0.11 (0.02) & 0.82 (0.02) & 0.61 (0.03) & 0.82 (0.04) & 0.17 (0.03) & 0.86 (0.03) & 0.89 (0.02) & 0.25 (0.02) & 0.22 (0.01) & 0.95 (0.01) & 0.11 (0.01) \\\\\n\\midrule\n60 & 250 & 50 & 0.70 (0.02) & 0.15 (0.02) & 0.77 (0.01) & 0.59 (0.04) & 0.82 (0.04) & 0.16 (0.03) & - & - & - & 0.35 (0.02) & 0.85 (0.02) & 0.21 (0.01) \\\\\n60 & 250 & 100 & 0.70 (0.01) & 0.13 (0.10) & 0.79 (0.05) & 0.62 (0.04) & 0.80 (0.04) & 0.17 (0.03) & - & - & - & 0.26 (0.01) & 0.89 (0.02) & 0.13 (0.01) \\\\\n60 & 250 & 200 & 0.76 (0.02) & 0.11 (0.01) & 0.85 (0.01) & 0.69 (0.05) & 0.80 (0.03) & 0.19 (0.04) & 0.91 (0.02) & 0.85 (0.02) & 0.33 (0.01) & 0.17 (0.01) & 0.84 (0.05) & 0.15 (0.03) \\\\\n\\midrule\n90 & 125 & 50 & 0.63 (0.04) & 0.10 (0.04) & 0.75 (0.03) & 0.52 (0.02) & 0.89 (0.03) & 0.10 (0.03) & - & - & - & 0.23 (0.01) & 0.88 (0.00) & 0.15 (0.01) \\\\\n90 & 125 & 100 & 0.66 (0.03) & 0.12 (0.03) & 0.74 (0.02) & 0.55 (0.04) & 0.87 (0.03) & 0.15 (0.02) & - & - & - & 0.18 (0.01) & 0.92 (0.01) & 0.10 (0.01) \\\\\n90 & 125 & 200 & 0.67 (0.02) & 0.13 (0.02) & 0.76 (0.01) & 0.57 (0.03) & 0.85 (0.04) & 0.17 (0.03) & - & - & - & 0.17 (0.01) & 0.94 (0.01) & 0.08 (0.01) \\\\\n\\midrule\n90 & 250 & 50 & 0.58 (0.03) & 0.09 (0.02) & 0.68 (0.03) & 0.54 (0.05) & 0.87 (0.04) & 0.11 (0.04) & - & - & - & 0.32 (0.01) & 0.84 (0.00) & 0.21 (0.01) \\\\\n90 & 250 & 100 & 0.65 (0.05) & 0.13 (0.04) & 0.73 (0.04) & 0.58 (0.04) & 0.82 (0.03) & 0.15 (0.03) & - & - & - & 0.18 (0.02) & 0.93 (0.01) & 0.11 (0.01) \\\\\n90 & 250 & 200 & 0.70 (0.02) & 0.12 (0.02) & 0.78 (0.01) & 0.61 (0.06) & 0.80 (0.04) & 0.18 (0.05) & - & - & - & 0.22 (0.01) & 0.89 (0.01) & 0.16 (0.01) \\\\\n\\bottomrule\t\t\t\t\t\n\\end{tabular}}\n\\label{s1}\n\\end{table}\n\nThe proposed FLiNG-BN has a few hyperparameters $L$, $M$, $\\alpha$, $(a_r, b_r)$, $(a_\\gamma, b_\\gamma)$, $(a_\\tau, b_\\tau)$, and $(a_\\sigma, b_\\sigma)$. We performed sensitivity analyses of these parameters at four different values with $(n, p, d) = (100, 30, 250)$. Results are summarized in Section C of the Supplementary Material. Our model appeared to be relatively robust within the tested ranges of hyperparameters.\n\n\\section{Applications} \\label{sec::eeg}\n\nWe applied the proposed FLiNG-BN to the brain EEG dataset downloaded from \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/eeg+database} \\citep{zhang1995event}. The dataset consists of 122 subjects with 77 in the alcoholic group and 45 in the control group, and was previously used to demonstrate functional undirected graphical models by \\cite{zhu2016bayesian} and \\cite{qiao2019functional}. The 64 electrodes placed on subjects' scalps (standard positions) measuring voltage values were sampled at 256 Hz for one second. Each subject completed 120 trials under one stimulus or two stimuli. See \\cite{zhang1995event} for details of the data collection procedure. We averaged all trials for each subject under the one stimulus condition. We separately analyzed these two groups to find their commonalities and differences of brain activity. Hence, we had $n = 77$ or $n = 45$ subjects and $p = 64$ functions representing the brain EEG signals at different scalp positions recorded at $d = 256$ time points. We focused on EEG signals filtered at $\\alpha$ frequency bands between 8 and 12.5 Hz using the \\texttt{eegfilt} function in the EEGLAB toolbox from MATLAB \\citep{delorme2004eeglab}.\n\nTo check the Gaussianity of the observed functions, we performed Shapiro--Wilk normality test \\citep{shapiro1965analysis} to each of $p = 64$ scalp positions at each of $d = 256$ time points. The null hypothesis (i.e., the observations are marginally Gaussian) was rejected for many combinations of scalp position and time point and therefore, the non-Gaussianity of the proposed model is deemed appropriate.\n\nFive orthonormal basis functions were selected for both the alcoholic and control group according to the procedure described in Section B of the Supplementary Material. We ran MCMC for 10,000 iterations, discarded the first half as burn-in, and retained every 10th iteration after burn-in. The estimated basis functions are shown in Figure \\ref{pfunction}. As evident from the plot, they are very similar across the two groups. The causal networks estimated by thresholding the posterior probability of inclusion at 0.9 are shown in Figure \\ref{fnetwork}. The sparsity level is approximately 3.0\\% for the alcoholic group and 2.5\\% for the\ncontrol group. \n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}[ht]{1 \\textwidth}\n\\centering\n\\includegraphics[width = 1 \\textwidth]{FALCOHOL}\n\\caption{Alcoholic group.}\n\\end{subfigure}\n\\begin{subfigure}[ht]{1 \\textwidth}\n\\centering\n\\includegraphics[width = 1 \\textwidth]{FCONTROL}\n\\caption{Control group.}\n\\end{subfigure}\n\\caption{Estimated basis functions from brain EEG records that explained 90\\% of the variation.}\n\\label{pfunction}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width = 1 \\textwidth]{COMBINEAC}\n\\caption{Estimated causal brain networks from EEG records by FLiNG-BN with posterior probability of inclusion $\\geq 0.9$, separately for the alcoholic (left) and control (right) group.}\n\\label{fnetwork}\n\\end{figure}\n\nOur results reveal several interesting patterns. First, the connection is relatively dense in the frontal region for both groups. Second, the alcoholic group has more directed connections detected in the left temporal and occipital regions. Third, most brain locations tend to connect to adjacent positions, while distant locations are much less connected. Figure \\ref{dnetwork} shows the common and differential networks for the two groups, where a substantial connectivity difference is observed between the two groups.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width = 1 \\textwidth]{COMBINECD}\n\\caption{Common (left panel) and differential (right panel) connections for the two groups. Black arrows indicate common connections, red arrows indicate connections detected by the alcoholic group only, and green arrows indicate connections detected by the control group only. }\n\\label{dnetwork}\n\\end{figure}\n\nIn addition, we demonstrated the proposed FLiNG-BN model with an additional application to COVID-19 multivariate longitudinal data, which have unevenly spaced measurements, in Section D of the Supplementary Material.\n\n\\section{Discussion} \\label{sec::discussion}\n\nIn this paper, we have proposed a functional Bayesian network model for causal discovery from multivariate functional data. We have discussed in detail a specific case of functional Bayesian network, namely the functional linear non-Gaussian model, and proved the underlying causal structure is identifiable even if the functions are purely observational and observed with noises. A fully Bayesian inference procedure has been proposed to implement our framework. Through simulation studies and real data applications, we have demonstrated the ability of our model in causal discovery.\n\nWe briefly discuss several possible directions to extend our current work. First, we may replace the underlying DAG with cyclic graphs, chain graphs, or ancestral graphs for more general causal and conditional independence structures. We have chosen a linear non-Gaussian SEM on the basis coefficients but this model can be replaced with a nonlinear SEM. Second, instead of fixing the number of basis functions, one could resort to increasing shrinkage priors \\citep{bhattacharya2011sparse, legramanti2020bayesian} to adaptively truncate redundant components. Finally, since we have two groups of observations in the EEG application, it would be interesting to jointly estimate the brain networks or directly estimate the differential network.\n\n\\bibliographystyle{jasa} \n\\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}