On the spectrum of non-selfadjoint Schr\"odinger operators with compact resolvent

TL;DR

This paper characterizes the Schatten class of the compact resolvent for certain non-selfadjoint Schrödinger operators with magnetic fields and complex potentials, and proves the completeness or infinite discrete spectrum of their eigenfunctions.

Contribution

It determines the Schatten class for the resolvent of non-selfadjoint Schrödinger operators obtained via analytic dilation, extending spectral analysis in unbounded domains.

Findings

Identifies Schatten class for the resolvent of these operators.

Proves completeness or infinite discrete spectrum of eigenfunctions in physical models.

Abstract

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schr\"odinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by Physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.