Lieb-Thirring type bounds for perturbed Schr\"odinger operators with single-well potentials

TL;DR

This paper establishes bounds on how much the eigenvalues of a perturbed one-dimensional Schr"odinger operator with a single-well potential can deviate from the lowest eigenvalue, using a variation of the commutation method and explicit factorisations for special cases.

Contribution

It introduces a new upper bound on eigenvalue deviations for perturbed Schr"odinger operators with single-well potentials, extending previous methods with explicit bounds for specific potentials.

Findings

Derived an upper bound for eigenvalue shifts in one-dimensional Schr"odinger operators.

Applied the method to P"oschl-Teller and Coulomb potentials for improved bounds.

Utilized a variation of the commutation method and explicit factorisations.

Abstract

We prove an upper bound on the sum of the distances between the eigenvalues of a perturbed Schr\"odinger operator $H_0-V$ and the lowest eigenvalue of $H_0$. Our results hold for operators $H_0=-\Delta-V_0$ in one dimension with single-well potentials. We rely on a variation of the well-known commutation method. In the P\"oschl-Teller and Coulomb cases we are able to use the explicit factorisations to establish improved bounds.