Summation formula inequalities for eigenvalues of the perturbed harmonic oscillator

TL;DR

This paper establishes explicit inequalities for sums of eigenvalues of one-dimensional Schr"{o}dinger operators, particularly for the perturbed harmonic oscillator, connecting bounds to the trace formula as the spectrum is fully covered.

Contribution

It introduces new explicit inequalities for eigenvalue sums of the perturbed harmonic oscillator, linking them to the trace formula in the full spectrum limit.

Findings

Derived explicit eigenvalue sum inequalities for Schr"{o}dinger operators.

Connected eigenvalue bounds to the trace formula in the full spectrum limit.

Provided convergence results for the bounds as the spectrum is fully considered.

Abstract

We derive explicit inequalities for sums of eigenvalues of one-dimensional Schr\"{o}dinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit as the number of eigenvalues covers the whole spectrum.