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

Abstract

We consider a twisted Laplacian ∆ ν,µ on the n-complex space associated with the sub-Laplacian of the Heisenberg group C× ω C n realized as a central extension of the real Heisenberg group H 2n+1 . The main results to which is aimed this paper concern the spectral theory of ∆ Γ ν,µ when acting on some L 2 space of Γ-automorphic functions of biweight (ν, µ) associated to given cocompat discrete subgroup Γ of the additive group C n . We describe its spectrum proving a stability theorem. Using the Selberg's approach, we give the explicit dimension formula for the corresponding L 2 -eigenspaces. We also construct a concrete basis of such L 2 -eigenspaces.

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.…”