pinecone/core-2020-05-10-deduplication huggingface dataset:
The NLS limit for bosons in a quantum waveguide

We consider a system of $N$ bosons confined to a thin waveguide, i.e.\ to a
region of space within an $\varepsilon$-tube around a curve in $\mathbb{R}^3$.
We show that when taking simultaneously the NLS limit $N\to \infty$ and the
limit of strong confinement $\varepsilon\to 0$, the time-evolution of such a
system starting in a state close to a Bose-Einstein condensate is approximately
captured by a non-linear Schr\"odinger equation in one dimension. The strength
of the non-linearity in this Gross-Pitaevskii type equation depends on the
shape of the cross-section of the waveguide, while the "bending" and the
"twisting" of the waveguide contribute potential terms. Our analysis is based
on an approach to mean-field limits developed by Pickl.Comment: Final version to appear in Annales Henri Poincar
On the construction of composite Wannier functions

We give a constructive proof for the existence of an $N$-dimensional Bloch
basis which is both smooth (real analytic) and periodic with respect to its
$d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The
constructed Bloch basis is conjugation symmetric when the underlying projection
has this symmetry, hence the corresponding exponentially localized composite
Wannier functions are real. In the second part of the paper we show that by
adding a weak, globally bounded but not necessarily constant magnetic field,
the existence of a localized basis is preserved.Comment: 32 pages, to appear in Annales Henri Poincar
Continuity of the spectrum of a field of self-adjoint operators

Given a family of self-adjoint operators $(A_t)_{t\in T}$ indexed by a
parameter $t$ in some topological space $T$, necessary and sufficient
conditions are given for the spectrum $\sigma(A_t)$ to be Vietoris continuous
with respect to $t$. Equivalently the boundaries and the gap edges are
continuous in $t$. If $(T,d)$ is a complete metric space with metric $d$, these
conditions are extended to guarantee H\"older continuity of the spectral
boundaries and of the spectral gap edges. As a corollary, an upper bound is
provided for the size of closing gaps.Comment: 15 pages, 1 figur
Counting function of magnetic resonances for exterior problems

We study the asymptotic distribution of the resonances near the Landau levels
$\Lambda\_q =(2q+1)b$, $q \in \mathbb{N}$, of the Dirichlet (resp. Neumann,
resp. Robin) realization in the exterior of a compact domain of $\mathbb{R}^3$
of the 3D Schr{\"o}dinger operator with constant magnetic field of scalar
intensity $b\textgreater{}0$. We investigate the corresponding resonance
counting function and obtain the main asymptotic term. In particular, we prove
the accumulation of resonances at the Landau levels and the existence of
resonance free sectors. In some cases, it provides the discreteness of the set
of embedded eigenvalues near the Landau levels
Quantum many-body fluctuations around nonlinear Schr\"odinger dynamics

We consider the many body quantum dynamics of systems of bosons interacting
through a two-body potential $N^{3\beta-1} V (N^\beta x)$, scaling with the
number of particles $N$. For $0< \beta < 1$, we obtain a norm-approximation of
the evolution of an appropriate class of data on the Fock space. To this end,
we need to correct the evolution of the condensate described by the
one-particle nonlinear Schr\"odinger equation by means of a fluctuation
dynamics, governed by a quadratic generator.Comment: 78 pages, acknowledgements adde
Wedge-local fields in integrable models with bound states II. Diagonal
  S-matrix

We construct candidates for observables in wedge-shaped regions for a class
of 1+1-dimensional integrable quantum field theories with bound states whose
S-matrix is diagonal, by extending our previous methods for scalar S-matrices.
Examples include the Z(N)-Ising models, the A_N-affine Toda field theories and
some S-matrices with CDD factors.
  We show that these candidate operators which are associated with elementary
particles commute weakly on a dense domain. For the models with two species of
particles, we can take a larger domain of weak commutativity and give an
argument for the Reeh-Schlieder property.Comment: minor corrections, as to appear in Ann H Poincare; 46 pages, 2 tikz
  figure
On Schroedinger operators with inverse square potentials on the
  half-line

The paper is devoted to operators given formally by the expression
\begin{equation*} -\partial_x^2+\big(\alpha-\frac14\big)x^{-2}. \end{equation*}
This expression is homogeneous of degree minus 2. However, when we try to
realize it as a self-adjoint operator for real $\alpha$, or closed operator for
complex $\alpha$, we find that this homogeneity can be broken.
  This leads to a definition of two holomorphic families of closed operators on
$L^2({\mathbb R}_+)$, which we denote $H_{m,\kappa}$ and $H_0^\nu$, with
$m^2=\alpha$, $-1<\Re(m)<1$, and where $\kappa,\nu\in{\mathbb C}\cup\{\infty\}$
specify the boundary condition at $0$. We study these operators using their
explicit solvability in terms of Bessel-type functions and the Gamma function.
In particular, we show that their point spectrum has a curious shape: a string
of eigenvalues on a piece of a spiral. Their continuous spectrum is always
$[0,\infty[$. Restricted to their continuous spectrum, we diagonalize these
operators using a generalization of the Hankel transformation. We also study
their scattering theory.
  These operators are usually non-self-adjoint. Nevertheless, it is possible to
use concepts typical for the self-adjoint case to study them. Let us also
stress that $-1<\Re(m)<1$ is the maximal region of parameters for which the
operators $H_{m,\kappa}$ can be defined within the framework of the Hilbert
space $L^2({\mathbb R}_+)$.Comment: The title has been changed, the previous one was : On almost
  homogeneous Schroedinger operator
The generalised principle of perturbative agreement and the thermal mass

The Principle of Perturbative Agreement, as introduced by Hollands & Wald, is
a renormalisation condition in quantum field theory on curved spacetimes. This
principle states that the perturbative and exact constructions of a field
theoretic model given by the sum of a free and an exactly tractable interaction
Lagrangean should agree. We develop a proof of the validity of this principle
in the case of scalar fields and quadratic interactions without derivatives
which differs in strategy from the one given by Hollands & Wald for the case of
quadratic interactions encoding a change of metric. Thereby we profit from the
observation that, in the case of quadratic interactions, the composition of the
inverse classical M{\o}ller map and the quantum M{\o}ller map is a contraction
exponential of a particular type. Afterwards, we prove a generalisation of the
Principle of Perturbative Agreement and show that considering an arbitrary
quadratic contribution of a general interaction either as part of the free
theory or as part of the perturbation gives equivalent results. Motivated by
the thermal mass idea, we use our findings in order to extend the construction
of massive interacting thermal equilibrium states in Minkowski spacetime
developed by Fredenhagen & Lindner to the massless case. In passing, we also
prove a property of the construction of Fredenhagen & Lindner which was
conjectured by these authors.Comment: 57 pages; alternative PPA proof in Sec. 3.4 revised and given only
  for non-derivative perturbations; added Appendices C and D to substantiate
  certain statements in Sec. 5; v3: minor improvements in Lemma 3.2 & D.1,
  added Remark 5.
Poincar\'e theory for decomposable cofrontiers

We extend Poincar\'e's theory of orientation-preserving homeomorphisms from
the circle to circloids with decomposable boundary. As special cases, this
includes both decomposable cofrontiers and decomposable cobasin boundaries.
More precisely, we show that if the rotation number on an invariant circloid
$A$ of a surface homeomorphism is irrational and the boundary of $A$ is
decomposable, then the dynamics are monotonically semiconjugate to the
respective irrational rotation. This complements classical results by Barge and
Gillette on the equivalence between rational rotation numbers and the existence
of periodic orbits and yields a direct analogue to the Poincar\'e
Classification Theorem for circle homeomorphisms. Moreover, we show that the
semiconjugacy can be obtained as the composition of a monotone circle map with
a `universal factor map', only depending on the topological structure of the
circloid. This implies, in particular, that the monotone semiconjugacy is
unique up to post-composition with a rotation. If, in addition, $A$ is a
minimal set, then the semiconjugacy is almost one-to-one if and only if there
exists a biaccessible point. In this case, the dynamics on $A$ are almost
automorphic. Conversely, we use the Anosov-Katok method to build a
$C^\infty$-example where all fibres of the semiconjugacy are non-trivial.Comment: 20 pages, 2 figures. Updated version addressing comments by the
  refere
Non-Backtracking Loop Soups and Statistical Mechanics on Spin Networks

We introduce and study a Markov field on the edges of a graph $\mathcal{G}$ in dimension $\textit{d}$ ≥ 2 whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set that splits the graph into two parts, the distributions of the loops and arcs contained in the two parts are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension.The first author acknowledges the support of Vidi Grant 639.032.916 of the Netherlands Organization for Scientific Research (NWO). The second author was partially supported by the Knut and Alice Wallenberg Foundation.This is the final version of the article. It first appeared from Springer via https://doi.org/10.1007/s00023-016-0524-
Spectral theory for Schr\"odinger operators with $\delta$-interactions
  supported on curves in $\mathbb R^3$

The main objective of this paper is to systematically develop a spectral and
scattering theory for selfadjoint Schr\"odinger operators with
$\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide
bounds for the number of negative eigenvalues depending on the geometry of the
curve, prove an isoperimetric inequality for the principal eigenvalue, derive
Schatten--von Neumann properties for the resolvent difference with the free
Laplacian, and establish an explicit representation for the scattering matrix.Comment: to appear in Annales Henri Poincar
Poisson algebras for non-linear field theories in the Cahiers topos

We develop an approach to construct Poisson algebras for non-linear scalar
field theories that is based on the Cahiers topos model for synthetic
differential geometry. In this framework the solution space of the field
equation carries a natural smooth structure and, following Zuckerman's ideas,
we can endow it with a presymplectic current. We formulate the Hamiltonian
vector field equation in this setting and show that it selects a family of
observables which forms a Poisson algebra. Our approach provides a clean
splitting between geometric and algebraic aspects of the construction of a
Poisson algebra, which are sufficient to guarantee existence, and analytical
aspects that are crucial to analyze its properties.Comment: v2: 24 pages; compatible with version to appear in Annales Henri
  Poincar
On the $L^p$ boundedness of wave operators for four-dimensional
  Schr\"odinger Operators with a threshold eigenvalue

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^4)$ with
real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient
pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty}
e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^4)$ for all $1\leq
p\leq \infty$ if zero is not an eigenvalue or resonance, and on $\frac43<p<4$
if zero is an eigenvalue but not a resonance. We show that in the latter case,
the wave operators are also bounded on $L^p(\mathbb R^4)$ for $1\leq p\leq
\frac43$ by direct examination of the integral kernel of the leading terms.
Furthermore, if $\int_{\mathbb R^4} xV(x) \psi(x) \, dx=0$ for all zero energy
eigenfunctions $\psi$, then the wave operators are bounded on $L^p$ for $1 \leq
p<\infty$.Comment: Updated references and made changes according to referee suggestions.
  To appear in Annales Henri Poincare, 20 page
3d Quantum Gravity: Coarse-Graining and q-Deformation

The Ponzano-Regge state-sum model provides a quantization of 3d gravity as a
spin foam, providing a quantum amplitude to each 3d triangulation defined in
terms of the 6j-symbol (from the spin-recoupling theory of SU(2)
representations). In this context, the invariance of the 6j-symbol under 4-1
Pachner moves, mathematically defined by the Biedenharn-Elliot identity, can be
understood as the invariance of the Ponzano-Regge model under coarse-graining
or equivalently as the invariance of the amplitudes under the Hamiltonian
constraints. Here we look at length and volume insertions in the
Biedenharn-Elliot identity for the 6j-symbol, derived in some sense as higher
derivatives of the original formula. This gives the behavior of these
geometrical observables under coarse-graining. These new identities turn out to
be related to the Biedenharn-Elliot identity for the q-deformed 6j-symbol and
highlight that the q-deformation produces a cosmological constant term in the
Hamiltonian constraints of 3d quantum gravity.Comment: 18 page
Bifurcations of standing localized waves on periodic graphs

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric
graph subject to the Kirchhoff boundary conditions. Bifurcations of standing
localized waves for frequencies lying below the bottom of the linear spectrum
of the associated stationary Schrodinger equation are considered by using
analysis of two-dimensional discrete maps near hyperbolic fixed points. We
prove existence of two distinct families of small-amplitude standing localized
waves, which are symmetric about the two symmetry points of the periodic
graphs. We also prove properties of the two families, in particular, positivity
and exponential decay. The asymptotic reduction of the two-dimensional discrete
map to the stationary NLS equation on an infinite line is discussed in the
context of the homogenization of the NLS equation on the periodic metric graph.Comment: 23 page
Ancilla dimension in quantum channel discrimination

Single-shot quantum channel discrimination is a fundamental task in quantum
information theory. It is well known that entanglement with an ancillary system
can help in this task, and furthermore that an ancilla with the same dimension
as the input of the channels is always sufficient for optimal discrimination of
two channels. A natural question to ask is whether the same holds true for the
output dimension. That is, in cases when the output dimension of the channels
is (possibly much) smaller than the input dimension, is an ancilla with
dimension equal to the output dimension always sufficient for optimal
discrimination? We show that the answer to this question is "no" by
construction of a family of counterexamples. This family contains instances
with arbitrary finite gap between the input and output dimensions, and still
has the property that in every case, for optimal discrimination, it is
necessary to use an ancilla with dimension equal to that of the input.
  The proof relies on a characterization of all operators on the trace norm
unit sphere that maximize entanglement negativity. In the case of density
operators we generalize this characterization to a broad class of entanglement
measures, which we call weak entanglement measures. This characterization
allows us to conclude that a quantum channel is reversible if and only if it
preserves entanglement as measured by any weak entanglement measure, with the
structure of maximally entangled states being equivalent to the structure of
reversible maps via the Choi isomorphism. We also include alternate proofs of
other known characterizations of channel reversibility.Comment: v2: updated references and minor modifications; in Annales Henri
  Poincare, November 201
Dynamical crossing of an infinitely degenerate critical point

We study the evolution of a driven harmonic oscillator with a time-dependent
frequency $\omega_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a
point of infinite spectral degeneracy. If the system is initialized in the
instantaneous vacuum in the distant past then the asymptotic future state is a
squeezed state whose parameters are explicitly determined. We show that the
squeezing is independent on the sweeping rate. This manifests the failure of
the adiabatic approximation at points where infinitely many eigenvalues
collide. We extend our analysis to the situation where the gap at $t=0$ remains
finite. We also discuss the natural geometry of the manifold of squeezed
states. We show that it is realized by the Poincar\'e disk model viewed as a
K\"ahler manifold
The $K$-theoretic bulk-edge correspondence for topological insulators

We study the application of Kasparov theory to topological insulator systems
and the bulk-edge correspondence. We consider observable algebras as modelled
by crossed products, where bulk and edge systems may be linked by a short exact
sequence. We construct unbounded Kasparov modules encoding the dynamics of the
crossed product. We then link bulk and edge Kasparov modules using the Kasparov
product. Because of the anti-linear symmetries that occur in topological
insulator models, real $C^*$-algebras and $KKO$-theory must be used.Comment: Minor corrections, to appear in Annales Henri Poincar\'
Hydrodynamic limit equation for a lozenge tiling Glauber dynamics

We study a reversible continuous-time Markov dynamics on lozenge tilings of
the plane, introduced by Luby et al. Single updates consist in concatenations
of $n$ elementary lozenge rotations at adjacent vertices. The dynamics can also
be seen as a reversible stochastic interface evolution. When the update rate is
chosen proportional to $1/n$, the dynamics is known to enjoy especially nice
features: a certain Hamming distance between configurations contracts with time
on average and the relaxation time of the Markov chain is diffusive, growing
like the square of the diameter of the system. Here, we present another
remarkable feature of this dynamics, namely we derive, in the diffusive time
scale, a fully explicit hydrodynamic limit equation for the height function (in
the form of a non-linear parabolic PDE). While this equation cannot be written
as a gradient flow w.r.t. a surface energy functional, it has nice analytic
properties, for instance it contracts the $\mathbb L^2$ distance between
solutions. The mobility coefficient $\mu$ in the equation has non-trivial but
explicit dependence on the interface slope and, interestingly, is directly
related to the system's surface free energy. The derivation of the hydrodynamic
limit is not fully rigorous, in that it relies on an unproven assumption of
local equilibrium.Comment: 31 pages, 8 figures. v2: typos corrected, some proofs clarified. To
  appear on Annales Henri Poincar
Renormalization Group Analysis of the Hierarchical Anderson Model

We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical
Anderson model to establish a criterion on the single-site distribution which
ensures exponential dynamical localization as well as positive inverse
participation ratios and Poisson statistics of eigenvalues. Our criterion
applies to all cases of exponentially decaying hierarchical hopping strengths
and holds even for spectral dimension $d > 2$, which corresponds to the regime
of transience of the underlying hierarchical random walk. This challenges
recent numerical findings that the spectral dimension is significant as far as
the Anderson transition is concerned
Monotonicity of the Quantum Relative Entropy Under Positive Maps

We prove that the quantum relative entropy decreases monotonically under the
application of any positive trace-preserving linear map, for underlying
separable Hilbert spaces. This answers in the affirmative a natural question
that has been open for a long time, as monotonicity had previously only been
shown to hold under additional assumptions, such as complete positivity or
Schwarz-positivity of the adjoint map. The first step in our proof is to show
monotonicity of the sandwiched Renyi divergences under positive
trace-preserving maps, extending a proof of the data processing inequality by
Beigi [J. Math. Phys. 54, 122202 (2013)] that is based on complex interpolation
techniques. Our result calls into question several measures of non-Markovianity
that have been proposed, as these would assess all positive trace-preserving
time evolutions as Markovian.Comment: v3: published version; v2: 9 pages, extended results to infinite
  dimensions and lifted trace-preservation condition, added discussions and
  references; v1: 5 page
The energy of the alphabet model

We call \emph{Alphabet model} a generalization to N types of particles of the
classic ABC model. We have particles of different types stochastically evolving
on a one dimensional lattice with an exchange dynamics. The rates of exchange
are local but under suitable conditions the dynamics is reversible with a
Gibbsian like invariant measure with long range interactions. We discuss
geometrically the conditions of reversibility on a ring that correspond to a
gradient condition on the graph of configurations or equivalently to a
divergence free condition on a graph structure associated to the types of
particles. We show that much of the information on the interactions between
particles can be encoded in associated \emph{Tournaments} that are a special
class of oriented directed graphs. In particular we show that the interactions
of reversible models are corresponding to strongly connected tournaments. The
possible minimizers of the energies are in correspondence with the Hamiltonian
cycles of the tournaments. We can then determine how many and which are the
possible minimizers of the energy looking at the structure of the associated
tournament. As a byproduct we obtain a probabilistic proof of a classic Theorem
of Camion \cite{Camion} on the existence of Hamiltonian cycles for strongly
connected tournaments. Using these results we obtain in the case of an equal
number of k types of particles new representations of the Hamiltonians in terms
of translation invariant $k$-body long range interactions. We show that when
$k=3,4$ the minimizer of the energy is always unique up to translations.
Starting from the case $k=5$ it is possible to have more than one minimizer. In
particular it is possible to have minimizers for which particles of the same
type are not joined together in single clusters.Comment: Reorganized according to referees report, new example
Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick
  model with ferromagnetic interaction

We consider a spherical spin system with pure 2-spin spherical
Sherrington-Kirkpatrick Hamiltonian with ferromagnetic Curie-Weiss interaction.
The system shows a two-dimensional phase transition with respect to the
temperature and the coupling constant. We compute the limiting distributions of
the free energy for all parameters away from the critical values. The zero
temperature case corresponds to the well-known phase transition of the largest
eigenvalue of a rank 1 spiked random symmetric matrix. As an intermediate step,
we establish a central limit theorem for the linear statistics of rank 1 spiked
random symmetric matrices.Comment: 45 pages, references adde