Summation formula inequalities for eigenvalues of Schr\"odinger operators

TL;DR

This paper develops inequalities for sums of eigenvalues of Schrödinger operators on finite intervals and tori, connecting these inequalities to classical trace formulas as the number of eigenvalues increases.

Contribution

It introduces new inequalities for eigenvalue sums of Schrödinger operators that converge to classical trace formulas in the limit.

Findings

Derived inequalities for eigenvalue sums on finite intervals and tori.

Established convergence of these inequalities to classical trace formulas.

Provided a unified approach linking eigenvalue inequalities and trace formulas.

Abstract

We derive inequalities for sums of eigenvalues of Schr\"{o}dinger operators on finite intervals and tori. In the first of these cases, the inequalities converge to the classical trace formulae in the limit as the number of eigenvalues considered approaches infinity.