Number of Eigenvalues of Non-self-adjoint Schr\"{o}dinger Operators with Dilation Analytic Complex Potentials

TL;DR

This paper establishes Lieb-Thirring inequalities for eigenvalues of non-self-adjoint Schrödinger operators with complex, dilation-analytic potentials, showing eigenvalues are invariant under complex dilation.

Contribution

It introduces Lieb-Thirring type bounds for complex potentials and proves eigenvalue invariance under complex dilation for non-self-adjoint Schrödinger operators.

Findings

Derived Lieb-Thirring inequalities for complex potentials

Proved invariance of eigenvalues under complex dilation

Extended spectral analysis to non-self-adjoint operators

Abstract

In this paper, we give Lieb-Thirring type inequalities for isolated eigenvalues of $d$-dimensional non-selfadjoint Schr\"{o}dinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that isolated eigenvalues and their multiplicities are invariant under complex dilation.