Research Article | Open Access
On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group
We consider the special magnetic Laplacian given by . We show that is connected to the sub-Laplacian of a group of Heisenberg type given by realized as a central extension of the real Heisenberg group . We also discuss invariance properties of and give some of their explicit spectral properties.
The scaled Landau Hamiltonian (special Hermite operator)where is the Euler operator and is its complex conjugate, describes the quantum behavior of a charged, spinless particle on the configuration space 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 ) 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  and the rich list of references therein.
In the present paper, we study the spectral properties of the second-order differential operatoracting on the free Hilbert space . The parameters and are assumed to be real such that . The particular case of leads to . Thus, the Laplacian can be seen as the Landau Hamiltonian perturbed by the first-order differential operator In fact we haveBoth and correspond to an isotropic magnetic field of constant strength . However, is associated with a specific magnetic vector potential (see (38)) issued from the symmetric gauge by a special gauge involving the parameter . This gauge is intimately connected to . Geometrically, as is the case for (see [9, 10]), represents a Bochner Laplacian on the smooth sections of a Hermitian line bundle with connection over the manifold .
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 on the real line (via the so-called Segal-Bargmann transform). One can also realize it as -eigenspace associated with the lowest Landau level of the Landau Hamiltonian . The same observation holds when dealing with the so-called holomorphic automorphic functions. In advantage, such Landau Hamiltonian leaves invariant such space . 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 Hamiltonian given by (2).
The main results to which is aimed this paper concern the realization of 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 is also established (see Section 3). This group is realized as a central extension of the standard Heisenberg group . In this new group, the symplectic form is extended and replaced by a Hermitian product (details in Section 2). Invariance properties of are discussed in Section 3 and concrete description of its -spectral analysis is presented in Section 5. More precisely, we show that the spectrum is discrete and independent of the parameter and coincides with the Landau energy levels of the Landau Hamiltonian in (1). Moreover, each eigenvalue occurs with infinite multiplicity. An orthogonal basis for every -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 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 Group as a Central Extension of the Heisenberg Group
In this section, we follow the exposition given in  in order to realize as a central extension of the Heisenberg group , where denotes the standard Hermitian form on . Indeed, if and are two abelian groups and is a given mapping, then we define on the -law byWe say that is a central extension of by associated with if the short sequenceis exact and such that is in , the center of the group . This holds if one of the following two equivalent assertions is satisfied: (i) preserves the neutral element and verifies the cocycle relation For every .(ii) is a group.
Now, let be the real -plane identified with the complex plane and denotes the complex -space endowed with its standard Hermitian form for and in . We define to be the set endowed with the -law given byUnder (9), is a noncommutative nilpotent group of step two with center . The identity element is and the symmetric element of given is . Notice for instance that the -law given by (9) can be rewritten in the coordinates , , and as follows:Hence, endowing the set with the -law, given bymakes a group, which is nothing else than the classical real Heisenberg group of dimension . One can notice easily that , in addition of being the central extension of by associated with the map , can also be viewed, due to (10), as the central extension of by associated with . This can be stated otherwise using directly the following definition: the projection mapping , from onto , is a homomorphism from the group onto the Heisenberg group . Moreover, the kernel of is given by and is clearly contained in the center . Thus, we may say that the group is a central extension of the Heisenberg group by ; that is, we have . Accordingly, harmonic analysis on our group will have many links to that on the classical Heisenberg group.
3. Explicit Formula for the Sub-Laplacian on
The group with the -law given in (9) is a real Lie group of dimension , and its tangent space at its neutral element is given by viewed as a real vector space of dimension . In fact, is naturally equipped with the standard differentiable structure on Euclidean spaces, generated by the coordinates system , where is the coordinates mapThe group action and the group symmetric maps are smooth under this differentiable structure. Denote by its associated Lie algebra composed of all left-invariant vector fields on and endowed with the standard bracket on vector fields. For the sake of giving the explicit formula for the sub-Laplacian on , a basis of formed by first-order differential operators on functions of is needed. Define the left action by a fixed element byThis 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 is given explicitly by for test data and a smooth function of . By definition, a vector field is said to be left-invariant if the equality 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 . Recall that a basis of the tangent vector space acting on smooth functions is given bywhere is the ordinary partial derivative with respect to the th variable. We can now carry out the following computation in order to find generators for : We plug in in the middle of the last equation and we use the multivariable chain rule to get where and is the th coordinate map of . Explicitly, we have Therefore, the 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 and withfor . We summarize the above discussion on and its associated Lie algebra in the following statement.
Proposition 1. The real vector fields together with , ; given by form a basis for . Moreover, they satisfy the following commutation relations of Heisenberg type: for all .
Remark 2. As expected we see, in view of the above proposition, that the Lie algebra of with is also a central extension of the classical Heisenberg algebra generated by the vector fields with the nontrivial commutation relation , where ; , are the coordinates of .
Remark 3. To build such left-invariant vector fields, one can also look for a one parameter group of , that is, a group homomorphism satisfying
According to the above discussion, we can introduce the following definition of sub-Laplacian on .
Definition 4. Let ; , be the vector fields given in Proposition 1. Then, the operator is called a sub-Laplacian of .
The following proposition gives the explicit differential expression of in terms of the Laplace-Beltarmi of and the first-order differential operators and defined by Namely, we have the following result.
Proposition 5. The sub-Laplacian prescribed in Definition 4 is given explicitly in the coordinates , , of as follows:where and .
Proof. The explicit expression of given in Proposition 5 can be handled by straightforward computations.
Remark 6. If we consider the coordinates and with , then the sub-Laplacian in (31) can be rewritten aswhere is the complex Euler operator and is its complex conjugate.
Remark 7. The action of on functions on that are independent of the argument reduces to that of the sub-Laplacianof the classical Heisenberg group .
We conclude this section by mentioning that both operators and 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 operatorFormally, is related to using partial Fourier transform in with as dual arguments.
In the next section, we will prove that the operator can also be regarded as a Schrödinger operator in the presence of a uniform magnetic field on associated with a specific differential 1-form .
4. Realization of as a Magnetic Schrödinger Operator and Invariance Property
A magnetic Schrödinger operator on a complete oriented Riemannian manifold on scalar functions is in general of the formwhere is a given real differential -form on (magnetic vector potential). Here stands for the usual exterior derivative acting on the space of differential -forms , is the operator of exterior left multiplication by , that is, , and is the formal adjoint of with respect to the Hermitian product on induced by the metric , where denotes the Hodge star operator associated with the volume form. From general theory of Schrödinger operators on noncompact manifold (see for example ), it is known that the operator , viewed as an unbounded operator in , is essentially self-adjoint for any smooth measure .
In our framework is the complex -space equipped with its Kähler metric and the corresponding volume form is . Associated with the parameters and , we consider the potential vectorSome 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-valued function on , we have
Proof (sketched). We start by writing as Next, using the well-known facts and , we establish the following
One of the advantages of the realization of as (39), with the differential -form in (38), is that we can derive easily some invariance properties with respect to the group of rigid motions of the complex Hermitian space ; . Thus, let denote the group of biholomorphic mapping of that preserve the Hermitian metric . Then, is the group of semidirect product of the additive group with the unitary group of and can be represented asIt acts transitively on via the mappings . The pull-back of the differential -form by the above mapping ; is related to by the following identity.
Proposition 9. Let be as in (38). Then, for every we havewhereThe phase function is given by
Proof. The identity (43) holds by component-wise straightforward computations. Indeed, direct computation yields where is the inverse mapping of and for . Thus, the result follows thanks to .
Notice that the relation (43) reads also as and shows that the differential -form is not -invariant. But and 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 invariant with respect to a -action on functions built with the help of the following automorphic factor defined through (44) and satisfying the chain rulefor every and . Associated with , we define to be the operator acting on differential -forms of through the formula On -complex-valued functions on , it reduces further toThus, the following invariance property for holds.
Proposition 10. For every , we have
Proof. Using the well-known facts and , we get Now, by means of the identity (43), it follows thatMoreover, commutes also with for being a unitary transformation. Therefore, by means of the expression of as a magnetic Schrödinger operator , we deduce easily that and commute. This ends the proof.
Remark 11. For , where denotes the identity matrix in , the unitary operators given in (49) define projective representations of on the space of -functions on . 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 vector given through (38) can be seen as a specific magnetic vector potential that corresponds to an isotropic magnetic field of constant strength , since . It is issued from the holomorphic gauge and the symmetric gauge defined, respectively, by More exactly, we have the gauge transformations
Remark 13. The connection form in (38) readsin the real coordinates . Its divergence is then showed to be given by . Accordingly, corresponds to the constant term involved in the expression of the magnetic quantum Hamiltonian given through (34). Therefore, is not a radiation (coulomb) gauge, unless , and therefore can be seen as a perturbation of the Coulomb gauge. The perturbation operator in is given by so that .
5. Spectral Properties of Acting on and on
We denote by the Frechet space of complex-valued functions on endowed with the compact-open topology, while denotes the usual Hilbert space of square integrable complex-valued functions on with respect to the usual Lebesgue measure . In the sequel, we will give a concrete description of the eigenspaces of in both and . To this end, let be any complex number and denote by the eigenspace of corresponding to the eigenvalue in ; that is,Also, by we denote the subspace of whose elements satisfy . Namely, by elliptic regularity of , we have
The first result related to and is the following.
Proposition 14. The eigenspaces and are invariants under the -action given by (49), in the sense that for every we have
Proof. This can be handled easily making use of the invariance property (50) of by the unitary transformations .
Proposition 15. The set of spherical eigenfuctions of with as eigenvalue is a one-dimensional vector subspace of generated bywhere is denoting here the usual confluent hypergeometric function
Remark 16. By a “spherical” (or radial here) eigenfuction of , we mean any -invariant function satisfying for all and .
Proof (sketched). To prove the statement, we write in the polar coordinates with and as where stands for the tangential component of . The eigenvalue problem for radial functions , with reduces to the differential equationNext, making use of the appropriate change of function , we see that the previous equation leads to the confluent hypergeometric differential equation [20, page 193]whose regular solution at is the confluent hypergeometric function .
Remark 17. The existence of the gradient of the function , in the gauge transformation , is equivalent to multiplying the eigenstates of the Landau Hamiltonian ; , by the phase factor . In fact, this follows by considering the similarity transformation generated by the unitary operator