Estimates for eigenvalues of Schr\"{o}dinger operators with complex-valued potentials

TL;DR

This paper develops new bounds for the eigenvalues of non-self-adjoint Schrödinger operators using $L_p$-norms of complex potentials, extending previous results and addressing conjectures by Laptev and Safronov.

Contribution

It introduces improved eigenvalue estimates for complex-valued potentials in multi-dimensional Schrödinger operators, including cases with slowly decaying potentials.

Findings

Extended and refined eigenvalue bounds for non-self-adjoint Schrödinger operators.

Addressed and discussed conjectures by Laptev and Safronov.

Included analysis of operators with slowly decaying potentials.

Abstract

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schr\"{o}dinger operators are obtained in terms of $L_{p}$-norms of the potentials. The results extend and improve those obtained previously. In particular, diverse versions of an assertion conjectured by Laptev and Safronov are discussed. Schr\"{o}dinger operators with slowly decaying potentials are also considered.