On concrete spectral properties of a twisted-Laplacian associated to a central extension of the real Heisenberg group

We consider the magnetic Laplacian $\Delta_{\nu,\mu}$ on $\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ \Delta_{\nu,\mu}= 4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial \overline{z_j}} +2i\nu (E+ \overline{E} +n) +2\mu (E- \overline{E} ) -(\nu^2+\mu^2)|z|^2. $$ We show that $\Delta_{\nu,\mu}$ is connected to the sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_\omega \mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. We also discuss invariance properties of $\Delta_{\nu,\mu}$ and give some of their explicit spectral properties.


INTRODUCTION
In the present paper we study the spectral properties of the second order differential operator acting on the free Hilbert space L 2 (C n , dm), where E is the Euler operator and E is its complex conjugate. The parameters ν and µ are assumed to be real and µ > 0. The particular case of ν = 0 and µ = 2b with n = 1 leads to minus four times the special Hermite operator ( [18,20]) Such operator is the Hamiltonian describing the quantum behavior of a charged particle on the configuration space C n under the influence of a constant magnetic field [1]. Geometrically, L b represents a Bochner Laplacians ∇ * ∇ on the smooth sections of a Hermitian line bundle with connection ∇ over the manifold M = C n [1,11].
The main results to which is aimed this paper concern the realisation of ∆ ν,µ as a magnetic Schrödinger operator associated to a specific potential vector (Section 4). The connection to the sub-Laplacian of a group of Heisenberg type given by C × ω C n is also established (see Section 3). The group C × ω C n is realized as a central extension N ω = (C × C n , · ω ) of the standard Heisenberg group H 2n+1 = (R × C n , · mω ). In this new group, the symplectic form is extended and replaced by an Hermitian product (details in Section 2). Invariance properties of ∆ ν,µ are discussed in Section 3 and concrete description of its L 2 -spectral analysis is presented in Section 5. In Section 6, we use the factorization method [8,14] to generate eigenfunctions of ∆ ν,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 [10,19].

THE GROUP N ω = C × ω C n AS A CENTRAL EXTENSION OF THE HEISENBERG GROUP
H 2n+1 = R × Imω C n We realize N ω := C × ω C n as a central extension of the Heisenberg group H 2n+1 := R × Imω C n , where ω(z, w) denotes the standard Hermitian form on C n . To this end, we follow the exposition given in [13]. Being indeed, if (K, •) and (G, ) are two abelian groups and ψ : K × K → G a given mapping. On G × K we define the · ψ -law by We say that G × ψ K is a central extension of (K, •) by (G, ) associated to ψ if the short sequence 0 → K → G × ψ K → G → 0 is 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, to wit i) ψ preserves the neutral element ψ(0 K , 0 K ) = 0 G and verifies the cocycle relation s, t ∈ R} and C n denotes the complex n-space endowed with its standard Hermitian form ω(z, w) := z, w = n ∑ j=1 z jwj for z = (z 1 , z 2 , · · · , z n ) and w = (w 1 , w 2 , · · · , w n ) in C n . Thus, we define N ω = C × ω C n to be the set C × C n endowed with the ω-law given by (z 0 ; z) · ω (w 0 ; w) = (z 0 + w 0 + z, w ; z + w).
Hence, endowing the set R t × C n with the · Imω -law given by (t; z) · Imω (t ; w) = (t + t + Im z, w ; z + w), (2.3) makes R × Imω C n a group, which is nothing else than the classical real Heisenberg group of dimension 2n + 1. One can notice easily that (C × C n , · ω ), in addition of being the central extension of C n by C associated to the map ψ = ω, can also be viewed, due to (2.2), as the central extension of (R t × C n , · mω ) by R s associated to ψ = ω. This can be stated otherwise using directly the definition; if we denote by q the projection mapping from (R s × R t ) × ω C n onto R t × Imω C n given by q(s, t; z) = (t; z), one check that the mapping q is a homomorphism from the group N ω = C × ω C n onto the Heisenberg group H 2n+1 = R × Imω C n and that the kernel of q is given by Since ker q is contained in the center Z(N ω ) of N ω , we may say that the group N ω = C × ω C n is a central extension of the Heisenberg group H 2n+1 = R × Imω C n by (R s , +); i.e we have C × ω C n / ker q = C × ω C n /R s = H 2n+1 . Accordingly, harmonic analysis on our group N ω = C × ω C n will have many links to that on the classical Heisenberg group.

EXPLICIT FORMULA FOR THE SUB-LAPLACIAN
The group N ω = C × ω C n with the ω-law given in (2.1) is a real Lie group of dimension 2n + 2, and its tangent space at its neutral element e = (0; 0) ∈ C × C n is given by T (0;0) N ω = (C, +) × (C n , +) as a real vector space of dimension 2n + 2. In fact, N ω is naturally equipped with the standard differentiable structure on euclidean spaces generated by the coordinates system {(C × C n , x)}, where x is the coordinates map x : C × C n −→ R 2n+2 ; (z 0 , z) −→ (s, t, x 1 , y 1 , x 2 , y 2 , · · · , x n , y n ).
The group action and the group symmetric maps are smooth under this differentiable structure. Let denote by n ω its associated Lie algebra composed of all left invariant vector fields on N ω and endowed with the standard bracket on vector fields. It is a well known fact that n ω ∼ = T (0;0) N ω . For the sack of giving the explicit formula for the sub-Laplacian L ω on N ω = C × ω C n , we need to build a basis of n ω which will be constructed as first order differential operators on functions of N ω . Define the left action by a fixed element (z 0 ; z) ∈ N ω by (z 0 ;z) : N ω −→ N ω ; (w 0 ; w) −→ (z 0 ; z) · ω (w 0 ; 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 for test data p and f such that p ∈ N ω and f is a smooth function of N ω . By definition, a vector field X is said to be left invariant if the equality g * 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 using (z 0 ,z) . Recall that a basis of the tangent vector space T (0;0) N ω acting on smooth functions f is given by where ∂ i is the ordinary partial derivative with respect to the i-th variable. We can now carry out the following computation in order to find generators for n ω : We plug in x −1 • x in the middle of the last equation and we use the multivariable chain rule to get Therefore, it follows that J m,i can be viewed as the components of the following Jacobian matrix Reading vertically, column by column, we find the following basis Note that we are using the coordinates z 0 = s + it and z j = x j + iy j with for j = 1, · · · , n. We summarize the above discussion on N ω = C × ω C n and its associated Lie algebra n ω with some additional remarks by making the following statement.
Proposition 3.1. The real vector fields S = ∂ ∂s , T = ∂ ∂t together with X j , Y j ; j = 1, · · · , n given by form a basis for n ω . Moreover, they satisfy the following commutation relations of Heisenberg type

Remark 3.2.
As expected we see, in view of the above proposition, that the Lie algebra n ω of N ω = C × ω C n with ω(z, w) = z, w is also a central extension of the classical Heisenberg algebra H 2n+1 = R × Imω C n generated by the vector fields

Remark 3.3.
To build such left invariant vector fields, one can also look for a one parameter group of N ω , i.e., a group homomorphism γ : Next, we define in below the sub-Laplacian by setting Definition 3.4. Let X j , Y j ; j = 1, · · · , n, be the vector fields given in Proposition 3.1. Then, the operator The following proposition gives the explicit differential expression of L ω in terms of the Laplace-Beltarmi ∆ R 2n of C n = R 2n and the first order differential operators E x,y and F x,y defined by Namely, we have Proposition 3.5. The sub-Laplacian L ω as defined in the above definition is given explicitly in the coordinates t, s, x j , y j , j = 1, · · · , n, of N ω = C × ω C n as follows The explicit expression of L ω given in Proposition 3.5 can be handled by straightforward computations.
Remark 3.6. If we consider the coordinates (s, t) ∈ R 2 = C and z = (z 1 , · · · , z n ) ∈ C n with z j = x j + iy j , then the sub-Laplacian L ω in (3.4) can be rewritten as is the complex Euler operator and E = n ∑ j=1z j ∂ ∂z j is its complex conjugate.
Remark 3.7. The action of L ω on functions F(t; z) on N ω = C × ω C n that are independent of the argument s, reduces to that of the sub-Laplacian of the classical Heisenberg group R × Imω C n = H 2n+1 .
We conclude this section by mentioning that both operators L ω and L Imω are not elliptic. But they are not far from being such in many aspects of their spectral theory. We will make this precise by discussing in a concrete manner the spectral eigenfunction problem on C n of the associated elliptic differential operator Formally, ∆ ν,µ is related to L ω using partial Fourier transform in (s, t) with (iν, iµ) as dual arguments.
In the next section, we see that the operator ∆ ν,µ can also be regarded as Schrödinger operator on C n = R 2n in the presence of a uniform magnetic field − → B µ = idθ ν,µ on C n = R 2n associated to a specific differential 1-form θ ν,µ .

PROPERTY
Magnetic Schrödinger operator on a complete oriented Riemannian manifold (M, g) is defined to be where θ is a given C 1 real differential 1-form on M (potential vector). Here d stands for the usual exterior derivative acting on the space of differential p-forms Ω p (M), ext θ is the operator of exterior left multiplication by θ, i.e., (ext θ)ω = θ ∧ ω and (d + extθ) * is the formal adjoint of d + extθ with respect to the Hermitian product α, β Ω p = M α ∧ β induced by the metric g on Ω p = Ω p (M), where denotes the Hodge star operator associated to the volume form. From general theory of Schrödinger operators on non-compact manifold M (see for example [17]), it is known that the operator H θ , viewed as an unbounded operator in L 2 (M, dm), is essentially self-adjoint for any smooth measure dm.
In our framework M is the complex n-space C n equipped with its Kähler metric dx j ⊗ dy j and the corresponding volume form is Vol = dx 1 dy 1 · · · dx n dy n . Associated to the parameters ν and µ, we consider the potential vector Thus, we prove the following result concerning the twisted Laplacian defined by (3.7).

Proposition 4.1. For every complex-valued
Sketched proof. We start by writing H θ ν,µ := (d + ext θ ν,µ ) * (d + ext θ ν,µ ) as Next, using the well-known facts d * = − d and (ext θ) * = ext θ , we establish the following One of the advantages of the formula for ∆ ν,µ as given by (4.3) with the differential 1form θ ν,µ in (4.2) is that we can derive easily some invariance properties of the Laplacian ∆ ν,µ with respect to the group of rigid motions of the complex Hermitian space (C n , ds 2 ); ds 2 = n ∑ j=1 dz j ⊗ dz j . Let G denote the group of biholomorphic mapping of C n that preserve the Hermitian metric ds 2 . Then, G = U(n) C n is the group of semi-direct product of the unitary group U(n) of C n with the additive group (C n , +). It can be represented as (4.4) G := U(n) C n = g = A b 0 1 =: [A, b]; A ∈ U(n), b ∈ C and acts transitively on C n via the mappings g.z = Az + b. The pull-back g * θ ν,µ of the differential 1-form θ ν,µ by the above mapping z −→ g.z is related to θ ν,µ by the following identity for every g ∈ G = U(n) C n .
The phase function φ ν,µ (g, z) is given by Proof. The identity (4.5) holds by component-wise straightforward computations. Indeed, direct computation yields Notice that the relation (4.5) reads also as g * θ ν,µ = θ ν,µ + d log(j ν,µ (γ, z)) and shows that the differential 1-form θ ν,µ is not G-invariant. But g * θ ν,µ and θ ν,µ are in the same class of the de Rham cohomology group. Also it gives insight how to make, in view of the expression (4.3), the Laplacian ∆ ν,µ invariant with respect to a G-action on functions built with the help of the following automorphic factor j ν,µ (g, z) defined through (4.6) and satisfying the chain rule (4.8) j ν,µ (gg , z) = j ν,µ (g, g z)j ν,µ (g , z) for every g ∈ G = U(n) C n and z ∈ C n . Associated to j ν,µ , we define T ν,µ g to be the operator acting on differential p-forms ω of C n through the formula Thus, the following invariance property for ∆ ν,µ holds. Proposition 4.3. For every g ∈ U(n) C n , we have Proof. Using the well-known facts g * d = dg * and g * (α ∧ β) = g * α ∧ g * β, we get Now, by means of the identity (4.5), it follows g commutes also with (d + ext θ ν,µ ) * for T ν,µ g being a unitary transformation. Therefore, by means of the expression of ∆ ν,µ = −(d + ext θ ν,µ ) * (d + ext θ ν,µ ) as a magnetic Schrödinger operator H θ ν,µ , we deduce easily that ∆ ν,µ and T ν,µ g commute. This ends the proof.
Remark 4.4. For g ∈ {I} C n = (C n , +), the unitary operators T ν,µ g given in (4.9) define projective representation of G on the space of C ∞ functions on C n . 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.

SPECTRAL PROPERTIES OF
We denote by C n the Frechet space of complex-valued functions on C ∞ (C n ) endowed with the compact-open topology, while L 2 (C n , dm) denotes the usual Hilbert space of square integrable complex-valued functions F(z) on C n with respect to the usual Lebesgue measure dm(z). In the sequel, we will give a concrete description of the eigenspaces of ∆ ν,µ in both C ∞ (C n ) and L 2 (C n , dm). To this end, let λ be any complex number in C and E λ (∆ ν,µ ) be the eigenspace of ∆ ν,µ corresponding to the eigenvalue −2µ(2λ Also, by A 2 λ (∆ ν,µ ) we denote the subspace of L 2 (C n , dm) whose elements F(z) satisfy ∆ ν,µ F = −2µ(2λ + n)F. Namely, by elliptic regularity of ∆ ν,µ , we have The first result related to E λ (∆ ν,µ ) and A 2 λ (∆ ν,µ ) is the following. Proposition 5.1. The eigenspaces E λ (∆ ν,µ ) and A 2 λ (∆ ν,µ ) are invariants under the T ν,µ -action given by (4.9).

Remark 5.4.
According to the proof of the previous result, we make the following key observation that can deserve as outline of the proofs of Proposition 5.2 and the assertions below. Indeed, the operators ∆ ν,µ and ∆ 0,µ are unitary equivalent in L 2 (C n , dm). More precisely, we have Accordingly, we claim the following Proposition 5.5. Let (ν, µ) ∈ R 2 with µ > 0 and λ ∈ C. Then, the eigenspace A 2 λ (∆ ν,µ ) as defined (5.2) is non-zero (Hilbert) space if and only if λ = l with l = 0, 1, 2, · · · , is a positive integer number. Moreover, the spaces A 2 l (∆ ν,µ ), l = 0, 1, 2, · · · , are pairwise orthogonal in L 2 (C n , dm) and we have the following orthogonal decomposition in Hilbertian subspaces Remark 5.6. The claim 5.5 asserts that the spectrum of ∆ ν,µ in L 2 (C n , dm) is purely discrete and each of its eigenvalue −2µ(2l + n), l ∈ Z + , is independent of ν and occurs with infinite degeneracy, i.e., the eigengspace A 2 l (∆ ν,µ ) in (5.2) is of infinite dimension. Proposition 5.7. Let (ν, µ) ∈ R 2 with µ > 0. For fixed l = 0, 1, 2, · · · , let P l be the orthogonal eigenprojector operator from L 2 (C n , dm) onto the eigenspace A 2 l (∆ ν,µ ) with −2µ(2l + n) as eigenvalue. Then the Schwartz kernel P ν,µ l (z, w) of the operator P l is given by the following explicit formula P ν,µ l (z, w) = ( µ π ) n (n − 1 + l)! (n − 1)!l! j ν,µ (z, w)e − µ 2 |z−w| 2 1 F 1 (−l; n; µ|z − w| 2 ), (5.4) where the factor j ν,µ (z, w); z, w ∈ C n is given by Sketched proof. The proof for ν = 0 is contained in [3,2,6]. For arbitrary ν, the proof can be handled in a similar way or making use of the key observation that in L 2 (C n , dm), the operators ∆ ν,µ and ∆ 0,µ are unitary equivalents and we have ∆ ν,µ = e − iν 2 |z| 2 ∆ 0,µ e + iν 6. FACTORISATION OF ∆ ν,µ AND ASSOCIATED HERMITE POLYNOMIALS In this section we study the spectral theory of ∆ ν,µ on L 2 (C n , dλ) using the factorisation method. This method finds its origin in the works of Dirac [4] and Schrödinger [16], then developed by Infeld and Hull [8] in order to solve eigenvalue problems appearing in quantum theory. Notice for instance that the operator ∆ 0,µ = L µ is refereed in physicmathematical literature as the Landau operator on R 2n = C n (or Schrödinger operator on R 2n in the presence of a uniform magnetic field − → B µ = √ −1dθ ν,µ ) and for which many of their spectral properties that we are considering go back to Landau's work in 1930 on the Hamiltonian in R 2 = C given by More generally the Laplacian ∆ ν,µ can be rewritten as Hence in view of the above remarks, the spectral properties of ∆ ν,µ on L 2 (C n , dm) or on C ∞ (C n ) can be derived from Landau's work [12]. To this end, It will be helpful to definẽ We also need to define, for j = 1, 2, · · · , n, the following first order differential operators