On a sum rule for Schr\"odinger operators with complex potentials

TL;DR

This paper investigates the eigenvalue distribution of one-dimensional Schrödinger operators with complex potentials, proving convergence of a series related to the eigenvalues' imaginary parts under certain decay conditions.

Contribution

It establishes a new sum rule for eigenvalues of Schrödinger operators with complex potentials, extending understanding of spectral properties with faster-than-Coulomb decay.

Findings

Series of imaginary parts of square roots of eigenvalues converges.

Convergence holds if potential decays faster than Coulomb potential.

Abstract

We study the distribution of eigenvalues of the one-dimensional Schr\"odinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then the series of imaginary parts of square roots of eigenvalues is convergent.