Eigenfunction Expansions of the Magnetic Schrödinger Operator in Bounded Domains.

Abstract

In this work, we introduce the magnetic Schrödinger operator corresponding to the generalized Dirichlet problem. We prove its self-adjointness and discreteness of the spectrum in bounded domains in the multidimensional case. We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space. We give a new characteristic of the definition domain of the magnetic Schrödinger operator. We investigate the existence and uniqueness of a solution of the magnetic Schrödinger equation with a spectral parameter. It is proved that if the spectral parameter is different from the eigenvalues, then the first generalized Dirichlet problem has a unique solution. We then find the solvability condition for the generalized Dirichlet problem when the spectral parameter coincides with the eigenvalue of the Schrödinger magnetic operator.