# On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group

Authors: Aymane El Fardi; Allal Ghanmi; Ahmed Intissar
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We consider the special magnetic Laplacian given byΔν,μ=4∑j=1n∂2/∂zj∂zj¯+2iνE+E¯+n+2μE-E¯-ν2+μ2z2. We show that Δν,μ is connected to the sub-Laplacian of a group of Heisenberg type given by C×ωCn realized as a central extension of the real Heisenberg group H2n+1. We also discuss invariance properties of Δν,μ and give some of their explicit spectral properties.


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## Body

## 1. Introduction

The scaled Landau Hamiltonian (special Hermite operator)(1)Hμ≔4∑j=1n∂2∂zj∂z¯j+2μE-E--μ2z2,where E is the Euler operator and E¯ is its complex conjugate, describes the quantum behavior of a charged, spinless particle on the configuration space Cn under the influence of a constant magnetic field. It has been considered and studied from different point of views in physics as in mathematics [1–7]. It goes back to L. D. Landau (for n=1) and plays an important role in many different contexts such as Feynman path integral (in Feynman-Kac formula), oscillatory stochastic integral, and theory of lattices electrons in uniform magnetic field. See Bellissard [8] and the rich list of references therein.In the present paper, we study the spectral properties of the second-order differential operator(2)Δν,μ=4∑j=1n∂2∂zj∂z¯j+2μ+iνE-2μ-iνE--ν2+μ2z2+2iνn,acting on the free Hilbert space L2Cn,dm. The parameters ν and μ are assumed to be real such that μ>0. The particular case of ν=0 leads to Δ0,μ=Hμ. Thus, the Laplacian Δν,μ can be seen as the Landau Hamiltonian Hμ perturbed by the first-order differential operator (3)Dν≔2iνE+E-+n-ν2z2.In fact we have(4)Δν,μ=Hμ+Dν.Both Δν,μ and Hμ correspond to an isotropic magnetic field of constant strength μ. However, Δν,μ is associated with a specific magnetic vector potential (see (38)) issued from the symmetric gauge by a special gauge involving the parameter ν. This gauge is intimately connected to Dν. Geometrically, as is the case for Hμ (see [9, 10]), Δν,μ represents a Bochner Laplacian on the smooth sections of a Hermitian line bundle with connection over the manifold M=Cn.The real motivation of considering such magnetic quantum Hamiltonian, and therefore the corresponding magnetic potential vector in (38), lies in the study of the space of biweighted holomorphic automorphic functions (defined by the means of the projective representation discussed in Section 4). Such space can be considered as the automorphic picture of the classical Bargmann’s space of entire functions which has physical implementation. In fact, such phase space is known to be unitary isomorphic to the quantum mechanical configuration space L2(R;dx) on the real line (via the so-called Segal-Bargmann transform). One can also realize it as L2-eigenspace associated with the lowest Landau level of the Landau Hamiltonian Hμ. The same observation holds when dealing with the so-called holomorphic automorphic functions. In advantage, such Landau Hamiltonian leaves invariant such space [11]. Accordingly, the first main object was the introduction of a special magnetic Schrödinger operator satisfying the following conditions:(i)
Leaving invariant the space of biweighted automorphic functions (not necessarily holomorphic).(ii)
The eigenspace associated with the lowest eigenvalue reduces to the space of biweighted holomorphic automorphic functions.The concrete construction gives rise to the magnetic quantum HamiltonianΔν,μ given by (2).The main results to which is aimed this paper concern the realization ofΔν,μ as a magnetic Schrödinger operator associated with a specific potential vector (Section 4). The connection to the sub-Laplacian of a group of Heisenberg type given by C×ωCn is also established (see Section 3). This group is realized as a central extension of the standard Heisenberg group H2n+1=(R×Cn,·Imω). In this new group, the symplectic form is extended and replaced by a Hermitian product (details in Section 2). Invariance properties of Δν,μ are discussed in Section 3 and concrete description of its L2-spectral analysis is presented in Section 5. More precisely, we show that the spectrum is discrete and independent of the parameter ν and coincides with the Landau energy levels of the Landau Hamiltonian Hμ in (1). Moreover, each eigenvalue occurs with infinite multiplicity. An orthogonal basis for every L2-eigenspace is next described and the corresponding reproducing kernel is given in a closed form. In Section 6, we use the factorization method [12–16] to generate eigenfunctions of Δν,μ in terms of multivariate version of complex Hermite polynomials. For the case of the twisted Laplacian of the standard Heisenberg group, one can refer to [6, 7, 17].


## 2. The GroupNω=C×ωCn as a Central Extension of the Heisenberg Group H2n+1=R×ImωCn

In this section, we follow the exposition given in [18] in order to realize Nω≔C×ωCn as a central extension of the Heisenberg group H2n+1≔R×ImωCn, where ωz,w denotes the standard Hermitian form on Cn. Indeed, if (K,•) and G,⊙ are two abelian groups and ψ:K×K→G is a given mapping, then we define on G×K the ·ψ-law by(5)z0;z·ψw0;w=z0⊙w0⊙ψz,w;z•w.We say that G×ψK is a central extension of (K,•) by (G,⊙) associated with ψ if the short sequence(6)0⟶K⟶G×ψK⟶G⟶0is exact and such that K is in Z(G), the center of the group E. This holds if one of the following two equivalent assertions is satisfied: (i)
ψ preserves the neutral element ψ(0K,0K)=0G and verifies the cocycle relation(7)ψx,y⊙ψx•y,z=ψx,y•z⊙ψy,z.For everyx,y,z∈K.(ii)
G×ψK≔(G×K,·ψ) is a group.Now, letR2=Rs×Rt be the real (s,t)-plane identified with the complex plane C=z0=s+it;s,t∈R and Cn denotes the complex n-space endowed with its standard Hermitian form (8)ωz,w≔z,w=∑j=1nzjw¯jfor z=z1,z2,…,zn and w=w1,w2,…,wn in Cn. We define Nω=C×ωCn to be the set C×Cn endowed with the ·ω-law given by(9)z0;z·ωw0;w=z0+w0+z,w;z+w.Under (9), Nω≔C×ωCn is a noncommutative nilpotent group of step two with center ZNω=C×ω0=R×R×ω0. The identity element is 0;0 and the symmetric element of given z0;z is -z0-z,z;-z. Notice for instance that the ·ω-law given by (9) can be rewritten in the coordinates z0=s,t, w0=s′,t′, and z,w∈Cn as follows:(10)s,t;z·ωs′,t′;w=s+s′+Rez,w,t+t′+Imz,w;z+w.Hence, endowing the set Rt×Cn with the ·Imω-law, given by(11)t;z·Imωt′;w=t+t′+Imz,w;z+w,makes R×ImωCn a group, which is nothing else than the classical real Heisenberg group of dimension 2n+1. One can notice easily that C×Cn,·ω, in addition of being the central extension of Cn by C associated with the map ψ=ω, can also be viewed, due to (10), as the central extension of Rt×Cn,·Imω by Rs associated with ψ=Reω. This can be stated otherwise using directly the following definition: the projection mapping qs,t;z=t;z, from Rs×Rt×ωCn onto Rt×ImωCn, is a homomorphism from the group Nω=C×ωCn onto the Heisenberg group H2n+1=R×ImωCn. Moreover, the kernel of q is given by (12)ker⁡q=s,0;0,s∈R=Rs×0×ω0and is clearly contained in the center Z(Nω). Thus, we may say that the group Nω=C×ωCn is a central extension of the Heisenberg group H2n+1=R×ImωCn by (Rs,+); that is, we have (C×ωCn)/ker⁡q=(C×ωCn/Rs)=H2n+1. Accordingly, harmonic analysis on our group Nω=C×ωCn will have many links to that on the classical Heisenberg group.


## 3. Explicit Formula for the Sub-LaplacianLω on Nω=C×ωCn

The groupNω=C×ωCn with the ·ω-law given in (9) is a real Lie group of dimension 2n+2, and its tangent space at its neutral element is given by T(0;0)Nω=C,+×Cn,+ viewed as a real vector space of dimension 2n+2. In fact, Nω is naturally equipped with the standard differentiable structure on Euclidean spaces, generated by the coordinates system (C×Cn,x), where x is the coordinates map(13)x:C×Cn⟶R2n+2;z0;z⟼s,t;x1,y1,x2,y2,…,xn,yn.The group action and the group symmetric maps are smooth under this differentiable structure. Denote by nω its associated Lie algebra composed of all left-invariant vector fields on Nω and endowed with the standard bracket on vector fields. For the sake of giving the explicit formula for the sub-Laplacian Lω on Nω=C×ωCn, a basis of nω formed by first-order differential operators on functions of Nω is needed. Define the left action by a fixed element (z0;z)∈Nω by(14)lz0;z:Nω⟶Nω;w0;w⟼z0;z·ωw0;w.This map is a diffeomorphism with respect to the Lie group structure. Hence, it is possible to extend its push-forward to act on vector fields. Furthermore, its action on a vector field X is given explicitly by (15)lg∗Xpf=Xpf∘lg,for test data p∈Nω and f a smooth function of Nω. By definition, a vector field X is said to be left-invariant if the equality lg∗X=X holds.In order to construct a left-invariant vector field basis, we take a basis of the tangent vectors at the identity and generate from each vector of the tangent basis a left-invariant vector field by pushing it forward usingl(z0,z). Recall that a basis of the tangent vector space T(0;0)Nω acting on smooth functions f is given by(16)∂∂xi0;0f≔∂if∘x-1x0,0;i=1,2,…,2n+2,where ∂i is the ordinary partial derivative with respect to the ith variable. We can now carry out the following computation in order to find generators for nω: (17)lz0;z∗∂∂xi0;0f=∂∂xi0;0f∘lz0;z=∂if∘lz0,z∘x-1x0,0.We plug in x-1∘x in the middle of the last equation and we use the multivariable chain rule to get (18)lz0;z∗∂∂xi0;0f=∂if∘x-1∘x∘lz0;z∘x-1x0,0=∑m=12n+2∂mf∘x-1x∘lz0;z∘x-1∘x0,0×∂ixm∘lz0;z∘x-1x0,0=∑m=12n+2Jm,i×∂∂xmz0;z,where Jm,i≔∂ixm∘lz0;z∘x-1x(0,0) and xm∘lz0;z∘x-1 is the mth coordinate map of x∘lz0;z∘x-1. Explicitly, we have (19)x∘lz0;z∘x-1s′,t′,x1′,y1′,…,xn′,yn′=s+s′+∑j=1nxjxj′+yjyj′,t+t′+∑j=1nyjxj′-xjyj′,x1+x1′,y1+y1′,…,yn+yn′.Therefore, the Jm,i can be viewed as the components of the following Jacobian matrix: (20)J≔J0,…,0=10x1y1⋯xnyn01y1-x1⋯yn-xn⋮0⋱0⋮⋱⋮⋱⋮⋱00⋯⋯1.Reading vertically, column by column, we find the following basis:(21)S=∂∂s,T=∂∂t,Xj=xj∂∂s+yj∂∂t+∂∂xj,Yj=yj∂∂s-xj∂∂t+∂∂yj.Note that we are using the coordinates z0=s+it and zj=xj+iyj with(22)∂∂x1=∂∂s,∂∂x2=∂∂t,∂∂x2j+1=∂∂xj,∂∂x2j+2=∂∂yj,for j=1,…,n. We summarize the above discussion on Nω=C×ωCn and its associated Lie algebra nω in the following statement.Proposition 1.
The real vector fields(23)S=∂∂s,T=∂∂ttogether with Xj, Yj; j=1,…,n given by (24)Xj=xj∂∂s+yj∂∂t+∂∂xj,Yj=yj∂∂s-xj∂∂t+∂∂yjform a basis for nω. Moreover, they satisfy the following commutation relations of Heisenberg type: (25)S,Xj=S,Yj=0,T,Xj=T,Yj=0,S,T=0,Xj,Xk=Yj,Yk=0,Xj,Yk=-2δjkTfor all j,k=1,…,n.Remark 2.
As expected we see, in view of the above proposition, that the Lie algebranω of Nω=C×ωCn with ωz,w=z,w is also a central extension of the classical Heisenberg algebra H2n+1=R×ImωCn generated by the vector fields (26)T=∂∂t,X~j=-yj∂∂t+∂∂xj,Y~j=xj∂∂t+∂∂yj;j=1,…,n,with the nontrivial commutation relation [X~j,Y~k]=-2T, where (xj,yj); j=1,…,n, are the coordinates of Cn=R2n.Remark 3.
To build such left-invariant vector fields, one can also look for a one parameter group ofNω, that is, a group homomorphism γ:(R,+)→Nω satisfying (27)γ˙0=dγdεεε=0=v0;v∈T0;0Nω=C×Cn.According to the above discussion, we can introduce the following definition of sub-Laplacian onNω=C×ωCn.Definition 4.
LetXj,Yj; j=1,…,n, be the vector fields given in Proposition 1. Then, the operator (28)Lω=∑j=1nXj2+Yj2is called a sub-Laplacian of Nω=C×ωCn.The following proposition gives the explicit differential expression ofLω in terms of the Laplace-Beltarmi ΔR2n of Cn=R2n(29)ΔR2n≔∑j=1n∂2∂xj2+∂2∂yj2and the first-order differential operators Ex,y and Fx,y defined by (30)Ex,y≔∑j=1nxj∂∂xj+yj∂∂yj,Fx,y≔∑j=1nxj∂∂yj-yj∂∂xj.Namely, we have the following result.Proposition 5.
The sub-LaplacianLω prescribed in Definition 4 is given explicitly in the coordinates t,s,xj,yj, j=1,…,n, of Nω=C×ωCn as follows:(31)Lω=ΔR2n+2Ex,y+n∂∂s-2Fx,y∂∂t+x2+y2∂2∂s2+∂2∂t2,where |x|2=∑j=1nxj2 and |y|2=∑j=1nyj2.Proof.
The explicit expression ofLω given in Proposition 5 can be handled by straightforward computations.Remark 6.
If we consider the coordinates(s,t)∈R2=C and z=z1,…,zn∈Cn with zj=xj+iyj, then the sub-Laplacian Lω in (31) can be rewritten as(32)Lω=4∑j=1n∂2∂zj∂zj¯+2E+E¯+n∂∂s-2iE-E¯∂∂t+z2∂2∂s2+∂2∂t2,where E=∑j=1nzj∂/∂zj is the complex Euler operator and E¯=∑j=1nzj¯∂/∂zj¯ is its complex conjugate.Remark 7.
The action ofLω on functions F(t;z) on Nω=C×ωCn that are independent of the argument s reduces to that of the sub-Laplacian(33)L~Imω=4∑j=1n∂2∂zj∂zj¯-2iE-E¯∂∂t+z2∂2∂t2of the classical Heisenberg group R×ImωCn=H2n+1.We conclude this section by mentioning that both operatorsLω and L~Imω are not elliptic. But they share many aspects and nice properties of their spectral theory with elliptic operators. We will precise this by giving the concrete description of the spectral eigenfunction problem of the associated elliptic differential operator(34)Δν,μ=4∑j=1n∂2∂zj∂zj¯+2μE-E¯+2iνE+E¯+n-ν2+μ2z2=4∑j=1n∂2∂zj∂zj¯+2μ+iνE-2μ-iνE¯-ν2+μ2z2+2iνn.Formally, Δν,μ is related to Lω using partial Fourier transform in (s,t) with (iν,iμ) as dual arguments.In the next section, we will prove that the operatorΔν,μ can also be regarded as a Schrödinger operator in the presence of a uniform magnetic field B→μ=idθν,μ on Cn=R2n associated with a specific differential 1-form θν,μ.


## 4. Realization ofΔν,μ as a Magnetic Schrödinger Operator and Invariance Property

A magnetic Schrödinger operator on a complete oriented Riemannian manifold(M,g) on scalar functions is in general of the form(35)Hθ=d+extθ∗d+extθ,where θ is a given C1 real differential 1-form on M (magnetic vector potential). Here d stands for the usual exterior derivative acting on the space of differential p-forms ΩpM, extθ is the operator of exterior left multiplication by θ, that is, extθω=θ∧ω, and d+extθ∗ is the formal adjoint of d+extθ with respect to the Hermitian product on Ωp=ΩpM(36)α,βΩp=∫Mα∧⋆βinduced by the metric g, where ⋆ denotes the Hodge star operator associated with the volume form. From general theory of Schrödinger operators on noncompact manifold M (see for example [19]), it is known that the operator Hθ, viewed as an unbounded operator in L2(M;dm), is essentially self-adjoint for any smooth measure dm.In our frameworkM is the complex n-space Cn equipped with its Kähler metric (37)ds2=-i2∑j=1ndzj⊗dz-j=∑j=1ndxj⊗dyjand the corresponding volume form is Vol(z)=dx1dy1⋯dxndyn. Associated with the parameters ν and μ, we consider the potential vector(38)θν,μz≔-μ-iν2∑j=1nzj¯dzj+μ+iν2∑j=1nzjdzj¯.Some comments on such vector potential are collected in Remarks 12, 13, and 17 below. Thus, we can prove the following result concerning the twisted Laplacian defined by (34).Proposition 8.
For every complex-valuedC∞ function f on Cn, we have(39)Δν,μf=-Hθν,μf=-d+extθν,μ∗d+extθν,μf.Proof (sketched).
We start by writingHθν,μ≔d+extθν,μ∗d+extθν,μ as (40)Hθν,μ=d∗df+d∗extθν,μf+extθν,μ∗df+extθν,μ∗extθν,μ.Next, using the well-known facts d∗=-⋆d⋆ and (extθ)∗=⋆extθ⋆, we establish the following (41)d∗d=-4∑j=1n∂∂zj∂∂zj¯,d∗extθν,μf+extθν,μ∗d=∑j=1n-2μ+iνzj∂∂zj+2μ-iνzj¯∂∂zj¯-2iν,extθν,μ∗extθν,μ=μ2+ν2z2.One of the advantages of the realization ofΔν,μ as (39), with the differential 1-form θν,μ in (38), is that we can derive easily some invariance properties with respect to the group of rigid motions of the complex Hermitian space Cn,ds2; ds2=∑j=1ndzj⊗dzj¯. Thus, let G denote the group of biholomorphic mapping of Cn that preserve the Hermitian metric ds2. Then, G=Cn⋊U(n) is the group of semidirect product of the additive group (Cn,+) with the unitary group U(n) of Cn and can be represented as(42)G≔Cn⋊Un=g=Ab01≕A,b;A∈Un,b∈Cn.It acts transitively on Cn via the mappings g·z=Az+b. The pull-back g∗θν,μ of the differential 1-form θν,μ by the above mapping z↦g; z is related to θν,μ by the following identity.Proposition 9.
Letθν,μ be as in (38). Then, for every g∈G=Cn⋊U(n) we have(43)g∗θν,μ=θν,μ+djν,μg,zjν,μg,z,where(44)jν,μg,z=exp⁡iϕν,μg,z.The phase function ϕν,μg,z is given by(45)ϕν,μg,z=-νRez,g-1·0+μImz,g-1·0.Proof.
The identity (43) holds by component-wise straightforward computations. Indeed, direct computation yields (46)g∗θν,μz=θν,μz-iν2dz,g-1·0+z,g-1·0¯+μ2dz,g-1·0-z,g-1·0¯=θν,μz+idϕν,μg,z,where g-1 is the inverse mapping of z↦g·z and g-1·0=-A-1b=-A∗b for g=[A,b]∈Cn⋊U(n). Thus, the result follows thanks to djν,μ(g,z)=id(ϕν,μ(g,z))jν,μ(g,z).Notice that the relation (43) reads also as g∗θν,μ=θν,μ+dlog⁡jν,μγ,z and shows that the differential 1-form θν,μ is not G-invariant. But g∗θν,μ and θν,μ are in the same class of the de Rham cohomology group. Also it gives insight how to make, in view of the expression (39), the Laplacian Δν,μ invariant with respect to a G-action on functions built with the help of the following automorphic factor jν,μg,z defined through (44) and satisfying the chain rule(47)jν,μgg′,z=jν,μg,g′zjν,μg′,zfor every g,g′∈G=Cn⋊U(n) and z∈Cn. Associated with jν,μ, we define Tgν,μ to be the operator acting on differential p-forms ω of Cn through the formula (48)Tgν,μω=jν,μg,zg∗ω.On C∞-complex-valued functions f on Cn, it reduces further to(49)Tgν,μfz=jν,μg,zg∗fz=jν,μg,zfg·z.Thus, the following invariance property for Δν,μ holds.Proposition 10.
For everyg∈Cn⋊U(n), we have(50)Δν,μTgν,μ=Tgν,μΔν,μ.Proof.
Using the well-known factsg∗d=dg∗ and g∗(α∧β)=g∗α∧g∗β, we get (51)Tgν,μd+extθν,μf=jν,μγ,zdg∗f+g∗θν,μ∧g∗f. Now, by means of the identity (43), it follows that(52)Tgν,μd+extθν,μf=jν,μγ,zdg∗f+jν,μγ,zθν,μg∗f+djν,μγ,zg∗f=djν,μγ,zg∗f+θν,μjν,μγ,zg∗f=d+extθν,μTgν,μf.Moreover, Tgν,μ commutes also with (d+extθν,μ)∗ for Tgν,μ being a unitary transformation. Therefore, by means of the expression of Δν,μ=-(d+extθν,μ)∗(d+extθν,μ) as a magnetic Schrödinger operator Hθν,μ, we deduce easily that Δν,μ and Tgν,μ commute. This ends the proof.Remark 11.
Forg∈Cn⋊{In}=(Cn,+), where In denotes the identity matrix in Cn×n, the unitary operators Tgν,μ given in (49) define projective representations of G on the space of C∞-functions on Cn. In fact, they are the so-called magnetic translation operators that arise in the study of Schrödinger operators in the presence of uniform magnetic field.Remark 12.
The potential vectorθν,μ given through (38) can be seen as a specific magnetic vector potential that corresponds to an isotropic magnetic field of constant strength 2μ, since d(iθν,μ)(z)=2μVol(z). It is issued from the holomorphic gauge θμh and the symmetric gauge θμs defined, respectively, by (53)θμh=-μ∑j=1nzj¯dzj,θμs=-μ2∑j=1nzj¯dzj-zjdzj¯.More exactly, we have the gauge transformations (54)θν,μ=θμh+iν+μ2d∑j=1nzj2,θν,μ=θμs+iν2d∑j=1zj2.Remark 13.
The connection formθν,μ in (38) reads(55)θν,μz=i∑j=1nμyj+νxjdxj-μxj-νyjdyjin the real coordinates x1,y1,x2,y2,…,xn;yn. Its divergence is then showed to be given by divθν,μ=2inν. Accordingly, divθν,μ corresponds to the constant term involved in the expression of the magnetic quantum Hamiltonian Δν,μ given through (34). Therefore, θν,μ is not a radiation (coulomb) gauge, unless ν=0, and therefore θν,μ can be seen as a perturbation of the Coulomb gauge. The perturbation operator in Δν,μ is given by (56)Dν≔2iνE+E-+n-ν2z2, so that Δν,μ=Hμ+Dν.


## 5. Spectral Properties ofΔν,μ Acting on C∞Cn and on H=L2Cn,dm

We denote byC∞(Cn) the Frechet space of complex-valued functions on Cn endowed with the compact-open topology, while L2(Cn,dm) denotes the usual Hilbert space of square integrable complex-valued functions F(z) on Cn with respect to the usual Lebesgue measure dm. In the sequel, we will give a concrete description of the eigenspaces of Δν,μ in both C∞(Cn) and L2(Cn,dm). To this end, let λ be any complex number and denote by Eλ(Δν,μ) the eigenspace of Δν,μ corresponding to the eigenvalue -2μ(2λ+n) in C∞(Cn); that is,(57)EλΔν,μ=F∈C∞Cn;Δν,μF=-2μ2λ+nF.Also, by Fλ2(Δν,μ) we denote the subspace of L2(Cn,dm) whose elements satisfy Δν,μF=-2μ(2λ+n)F. Namely, by elliptic regularity of Δν,μ, we have(58)Fλ2Δν,μ≔L2Cn,dm∩EλΔν,μ.The first result related toEλ(Δν,μ) and Fλ2(Δν,μ) is the following.Proposition 14.
The eigenspacesEλ(Δν,μ) and Fλ2(Δν,μ) are invariants under the Tν,μ-action given by (49), in the sense that for every g∈G=Cn⋊U(n) we have (59)Tgν,μEλΔν,μ⊂EλΔν,μ,Tgν,μFλ2Δν,μ⊂Fλ2Δν,μ.Proof.
This can be handled easily making use of the invariance property (50) of Δν,μ by the unitary transformations Tgν,μ.Proposition 15.
The set of spherical eigenfuctions ofΔν,μ with -2μ(2λ+n) as eigenvalue is a one-dimensional vector subspace of Eλ(Δν,μ) generated by(60)φλν,μz=e-μ-iν/2z2F11-λ;n;μz2,where 1F1(a;c;x) is denoting here the usual confluent hypergeometric function (61)F11a;c;x=1+acx1!+aa+1cc+1x22!+⋯;x∈C.Remark 16.
By a “spherical” (or radial here) eigenfuction ofΔν,μ, we mean any U(n)-invariant function f satisfying fh·z=f(z) for all h∈U(n) and z∈Cn.Proof (sketched).
To prove the statement, we writeΔν,μ in the polar coordinates z=rθ with r≥0 and θ∈S2n-1 as (62)Δν,μ=∂2∂r2+2n-1r+2iν∂∂r-ν2+μ2r2+2iνn+Lν,μθ, where Lν,μθ stands for the tangential component of Δν,μ. The eigenvalue problem Δν,μf=-2μ(2λ+n)f for radial functions f(z)=ψ(x), with x=r2 reduces to the differential equation(63)x∂2∂x2+n+iνx∂∂x-ν2+μ24x+iν-μ2n-μλψ=0.Next, making use of the appropriate change of function ψ(x)=eiν-μ/2xy(x), we see that the previous equation leads to the confluent hypergeometric differential equation [20, page 193](64)xy′′+n-μxy′+μλy=0whose regular solution at x=0 is the confluent hypergeometric function 1F1(-λ;n;μx).Remark 17.
The existence of the gradient of the functionφ(z)=iν/2|z|2, in the gauge transformation θν,μ=θμs+iν/2dz2, is equivalent to multiplying the eigenstates of the Landau Hamiltonian Hμ; ν=0, by the phase factor e-iν/2|z|2. In fact, this follows by considering the similarity transformation generated by the unitary operator Sf=e-iν/2|z|2f and next showing that(65)Δν,μ=e-iν/2z2Δ0,μe+iν/2z2.This is clear from the proof of Proposition 15. In other words the operators Δν,μ and Δ0,μ are unitary equivalent in L2(Cn,dm).Remark 18.
According to Remark17, any physical interpretation or application must take into account the parameter ν that is related to the special gauge transformation we have made.Remark 19.
The key observation is contained in the identity (65) which will serve as an outline of the proof of Proposition 15 as well as the proofs of the assertions below, taking into account the well-established results for Δ0,μ (see [1, 4–7] and the references therein).Accordingly, we claim the following.Proposition 20.
Let(ν,μ)∈R2 with μ>0 and λ∈C. Then, the eigenspace Fλ2(Δν,μ) as defined in (58) is nonzero (Hilbert) space if and only if λ=l with l=0,1,2,…, is a positive integer number. Moreover, the spaces Fl2(Δν,μ), l=0,1,2,…, are pairwise orthogonal in L2(Cn,dm) and we have the following orthogonal decomposition in Hilbertian subspaces: (66)L2Cn,dm=⨁l=0∞Fl2Δν,μ.Remark 21.
A direct proof of Proposition20 can be handled using Proposition 15 and the asymptotic behavior of the confluent hypergeometric function given by [20, page 332](67)F11a;c;x=Γc-x-aΓc-a+exxa-cΓa1+O1xas x→+∞. This asymptotic behavior can also be used to show that the radial function φλν,μ given by (60) is bounded if and only if λ=l; l=0,1,2,….Remark 22.
The unitary equivalence ofΔν,μ and Δ0,μ in L2(Cn,dm) shows that the spectrum of Δν,μ is purely discrete and coincides with the one of Δ0,μ. Thus, the energy levels (eigenvalues) are independent of the ν-parameter and are quantized as -2μ2l+n for varying l∈Z+. This is also contained in Proposition 20. Moreover, each energy level is infinitely degenerate, since it corresponds to infinite linearly independent states of the quantum system. The states ψp,q|l, given by (68) below, with index l constitute the lth Landau level.The following result prescribes the elements of theL2-eigenspaces of Δν,μ in terms of the confluent hypergeometric functions 1F1 and the harmonic polynomials hpq(z,z¯) on Cn that are homogeneous of degree p in z and degree q in z¯ (see [21]).Proposition 23.
An orthogonal basis of the infinite dimensional eigengspaceFl2(Δν,μ) in (58) is given by(68)ψp,q∣lμz=e-μ+iν/2z2F11q-l,n+p+q;μz2hpqz,z¯,for arbitrary nonnegative integers p,q with q≤l. Moreover, a function F belongs to Fl2(Δν,μ) if and only if it can be expanded as follows: (69)Fz=∑p=0+∞∑q=0lap,qψp,q∣lμz,where the constants ap,q satisfy the growth condition (70)FL2Cn;dm2=∑p=0+∞∑q=0lap,q21μn+p+ql-q!Γ2n+p+q2Γn+p+lhpqL2S2n-12<+∞.Proof.
The result follows immediately from the similarity transformation (65) combined with (ii) in Proposition  6 in [5].We conclude this section with the following assertion.Proposition 24.
Let(ν,μ)∈R2 with μ>0. For fixed l=0,1,2,…, let Pl be the orthogonal eigenprojector operator from L2(Cn,dm) onto the eigenspace Fl2(Δν,μ) with -2μ(2l+n) as eigenvalue. Then the Schwartz kernel Plν,μ(z,w) of the operator Pl is given by the following explicit formula:(71)Plν,μz,w=μπnn-1+l!n-1!l!jν,μz,we-μ/2z-w2F11-l;n;μz-w2,where the factor jν,μ(z,w), z,w∈Cn, is given by(72)jν,μz,w=e-iν/2z2-w2+μ/2z,w-z,w¯.Proof (sketched).
The proof forν=0 is contained in [1, 4, 5]. For arbitrary ν, the proof can be handled in a similar way or making use of the key observation that in L2(Cn,dm), the operators Δν,μ and Δ0,μ are unitary equivalents and we have (73)Δν,μ=e-iν/2z2Δ0,μe+iν/2z2.


## 6. Factorization ofΔν,μ and Associated Hermite Polynomials

In this section we study the spectral theory ofΔν,μ on L2(Cn,dλ) using the factorization method. This method finds its origin in the works of Dirac [12] and Schrödinger [13], then developed by Infeld and Hull [14] in order to solve eigenvalue problems appearing in quantum theory. For a recent survey see [16]. The key observation is that the Laplacian Δν,μ can be rewritten as (74)Δν,μ=-4∑j=1n-∂∂zj+μ-iν2zj¯∂∂zj¯+μ+iν2zj-2μnI.The spectral properties of Δν,μ on L2(Cn,dm) or on C∞(Cn), discussed in the previous section, can be derived from Landau’s work (see [22]). In the sequel, we will adopt the operator (75)Δ~ν,μ≔-14Δν,μ.The extra factor -1/4 is present to simplify later formulas. According to factorization above, we defined for every j=1,2,…,n, the following first-order differential operators(76)aj+=-∂∂zj+μ-iν2zj¯=-eμ-iν/2zj2∂∂zje-μ-iν/2zj2,(77)aj-=∂∂zj¯+μ+iν2zj=e-μ+iν/2zj2∂∂zj¯eμ+iν/2zj2.These operators satisfy the commutation relationshipsaj-,ak+=nμδj,k, where δj,k is the Krönecker symbol. They are linked to the Laplacian Δ~ν,μ through(78)∑j=1naj+aj-=Δ~ν,μ-n2μ,∑j=1naj-aj+=Δ~ν,μ+n2μ.Moreover, we have the following creation and annihilation identities: (79)Δ~ν,μaj+=aj+Δ~ν,μ+μ,Δ~ν,μaj-=aj-Δ~ν,μ-μand allow the determination of the eigenvalues and eigenvectors of Δ~ν,μ. Indeed, if ψ is an eigenvector of Δ~ν,μ associated with the eigenvalue λ we have the following:(80)Δ~ν,μaj+ψ=aj+Δ~ν,μ+μψ=λ+μaj+ψ,(81)Δ~ν,μaj-ψ=aj-Δ~ν,μ-μψ=λ-μaj-ψ.Thus, we need only to know those associated with the lowest eigenvalue. In fact, since Δ~ν,μ is positive semidefinite, all the eigenvalues are real and nonnegative. Moreover, from symmetry and ellipticity of Δ~ν,μ we know that Δ~ν,μ has an infinite sequence of nonnegative eigenvalues (see, e.g., [10]):(82)0≤λ0<λ1<⋯↑∞.Therefore, if ψ0 is an eigensolution associated with λ0, we have necessary aj-ψ0=0 for every j=1,2,…,n, thanks to (81) and therefore ψ0(z)=e-(μ+iν)/2z2f(z), where f is any arbitrary holomorphic function. This follows readily making use the expression of aj- in (77) involving the exponential. Consequently, (83)A0=keraj-=span¯zme-μ+iν/2z2;m∈Z+n,where zm for given multi-index m≔(m1,m2,…,mn) means zm≔z1m1z2m2⋯znmn. Making use of the creation operators we obtain the following family of multi-indexed functions hr,sν,μ; r=(r1,…,rn);s=(s1,…,sn):(84)hr,sν,μ=a1+r1⋯an+rnzse-μ+iν/2z2=-1reμ-iν/2z2Dzre-μ-iν/2z2zse-μ+iν/2z2=-1r+sμ-seμ-iν/2z2DzrDz¯se-μz2,where m and m! stand for m≔m1+⋯+m2 and m!≔m1!⋯mn!, respectively, and Dzm and Dz-m are defined by (85)Dzm≔∂m∂z1m1⋯∂znmn,Dz-m≔∂m∂z-1m1⋯∂z-nmn.According to the above discussion, hr,sν,μ are eigensolutions associated with the eigenvalue λl≔μn/2+l; l=r=0,1,2,…, of Δ~ν,μ. The following proposition shows that (86)λl≔μn2+l;l=0,1,2,…are the only eigenvalues of Δ~ν,μ.Proposition 25.
h
r
,
s
ν
,
μ form a complete orthogonal system in the Hilbert space L2(Cn,dλ). Moreover, we have the following decomposition L2(Cn,dλ)=⨁l=0∞Al, where (87)Al≔span¯hr,sν,μ;r,s∈Z+n,r=l.Proof.
The identity (84) shows that hr,sν,μ, up to eiν|z|2/2, are essentially the high-dimensional analogue of the univariate complex Hermite functions (88)hm,nσξ,ξ¯≔-1m+neσξ2∂m+n∂ξm∂ξ¯ne-σξ2;ξ∈C,σ>0,considered in [23, 24]. The main idea of the proof is then to separate the variable z in the expression (84) into its components zj to get (89)hr,sν,μz,z¯=μ-se-μ-iν/2z2∏j=1n-1rj+sjeμzj2∂rj+sj∂zjrj∂z¯jsje-μzj2=μ-se-μ-iν/2z2∏j=1nhrj,sjμzj,μz¯j.This implies that the eigenvalues ofΔν,μ=-4Δ~ν,μ are (90)-2μn+2l;l=0,1,2,…,which coincide with the results in Section 5. Notice as well that Fl2(Δν,μ) and Al refer to the same set.Remark 26.
The Hermite functionshr,sν,μ are given explicitly by (91)hr,sν,μ=μ-se-μ-iν/2z2∑k=0min⁡r,sμr+s-2k-1r-sr!s!k!r-k!s-k!zs-kz¯r-k,where r=(r1,…,rn), s=(s1,…,sn), and min⁡r,s≔(min⁡r1,s1,…,min⁡rn,sn).


## 7. Concluding Remarks

The consideration of the unitary transformationsTgν,μ, g∈G=Cn⋊U(n), and the Tgν,μ-invariance property satisfied by the magnetic Laplacian Δν,μ give rise to new class of biweighted automorphic functions associated with the automorphic factor jν,μ(g,z) when we restrict g to belong in a full-rank discrete subgroup Γ of G. We call them Γ-periodic functions of biweight (ν,μ). The considered Δν,μ leaves invariant this space and therefore the corresponding eigenvalue problem is well-defined. Thus, a detailed description of the spectral properties of Δν,μ when acting on biweighted automorphic functions with respect to any discrete subgroup of (Cn,+) (not necessary of full-rank) is of great interest. In this context, the particular case ν=0 and Γ=Cn⋊{1}, these functions reduce further the classical one studied in [11]. We hope to focus on this in a near future.



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