Lieb-Thirring and Jensen sums for non-self-adjoint Schr\"odinger operators on the half-line

TL;DR

This paper establishes optimal bounds for sums of eigenvalues of non-self-adjoint Schrödinger operators on the half-line, advancing understanding of spectral properties for these operators with various potentials.

Contribution

It provides the first comprehensive bounds for eigenvalue sums of non-self-adjoint Schrödinger operators, including both upper and lower bounds for different potential classes.

Findings

Upper bounds for eigenvalue sums are established for general integrable potentials.

Lower bounds are proven for specific potentials, showing optimality.

The results cover both critical and non-critical cases.

Abstract

We prove upper and lower bounds for sums of eigenvalues of Lieb-Thirring type for non-self-adjoint Schr\"odinger operators on the half-line. The upper bounds are established for general classes of integrable potentials and are shown to be optimal in various senses by proving the lower bounds for specific potentials. We consider sums that correspond to both the critical and non-critical cases.