Spectral properties of Landau Hamiltonians with non-local potentials

Abstract

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V, such that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ). First, we construct symbols V, possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ), and investigate its spectral asymptotics.