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

Abstract

We consider the special magnetic Laplacian given byshow that Δ ], is connected to the sub-Laplacian of a group of Heisenberg type given by C× C realized as a central extension of the real Heisenberg group 2 +1 . We also discuss invariance properties of Δ ], and give some of their explicit spectral properties.

Concrete $L^2$-spectral Analysis of a Bi-weighted $\Gamma$-automorphic Twisted Laplacian

“…From the general theory of Schrödinger operators on non-compact manifolds, the operator ∆ ν,µ , viewed as an unbounded operator in L 2 (C n ; dm), is essentially self-adjoint for any smooth measure dm (see for example [9]). It is proved in [4] that the L 2 -spectrum of ∆ ν,µ acting on the free Hilbert space L 2 (C n ; dm) is purely discrete, independent of the parameter ν and coincides with the Landau energy levels…”

“…is of infinite dimension. Moreover, the explicit expression of the reproducing kernel of the L 2 -eigenspaces F 2 (∆ ν,µ , C n ) in terms of the confluent hypergeometric function is given by [4,Proposition 5.11]…”

“…We conclude this section by providing another concrete geometrical realization of F ν,µ Γ (C n ) as equivariant functions of a rank-one principal bundle over C n /Γ. To this end, we invoke the group N ω := C × ω C n constructed in [4] and realized as a central extension of the Heisenberg group H 2n+1 := R × mω C n , where the mapping ω denotes the standard Hermitian form on C n given by ω(z, w) = z, w . Assume that we are given a lattice Γ 0 in (C, +) = (R 2 , +) and a lattice Γ in (C n , +) = (R 2n , +) such that ω sends Γ × Γ to Γ 0 , i.e., for every γ, γ , we have…”

“…The elliptic second order differential operator where ν and µ are real numbers such that µ > 0, has been considered recently in [4]. It is shown there that ∆ ν,µ is closely connected to the sub-Laplacian of a group of Heisenberg type C × ω C n using partial Fourier transform in (s, t) with (ν, µ) as dual arguments.…”

A lifting theorem for planar mixed automorphic functions and applications

“…However, we establish a lifting theorem, generalizing the particular one obtained in [4], and providing an isomorphic transformation between the eigenvalue problem of our constructed invariant Schrödinger operator on MAFs and the eigenvalue problem for the Landau Hamiltanian with constant magnetic field on the space of classical automorphic functions. As a side note, our considered class of MAFs are far to be confused with the ones defined in [2,3] or in [4,5]. Extension to the orbifold Γ\C n for given discrete subgroup Γ of U(n) ⋉ C n , acting on C n by the mappings g.…”