Weil-Titchmarsh theory as applied to the singular non-sectorial Schr\"odinger operator. Conditions for discreteness of the spectrum and compactness of the resolvent

TL;DR

This paper extends Weil-Titchmarsh theory to analyze the spectral properties of singular non-sectorial Schrödinger operators with complex potentials, providing conditions for discrete spectra and compact resolvents.

Contribution

It introduces new sufficient conditions for the discreteness of the spectrum and compactness of the resolvent for non-sectorial Schrödinger operators with complex potentials.

Findings

Provided criteria for spectrum discreteness

Established conditions for resolvent compactness

Extended spectral theory to broader class of operators

Abstract

The spectral properties of the singular Schr\"odinger operator with complex-valued potential which takes values in a wider region than the half-plane, have been little studied. In general case, the operator is non-sectorial, and the numerical range coincides with the entire complex plane. In this situation we propose sufficient conditions for discreteness of the spectrum and compactness of the resolvent.