Estimates for eigenvalues of the Schr\"odinger operator with a complex potential

TL;DR

This paper investigates the eigenvalue distribution of Schrödinger operators with complex potentials, showing that rapid decay of the potential confines all eigenvalues within a finite radius.

Contribution

It establishes conditions under which eigenvalues of Schrödinger operators with complex potentials are bounded, extending understanding of spectral properties with decaying potentials.

Findings

Eigenvalues are contained within a finite radius if the potential decays faster than Coulomb.

The decay rate of the potential influences the spectral bounds.

Provides new bounds on eigenvalue distribution for complex potentials.

Abstract

We study the distribution of eigenvalues of the Schr\"odinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.