Eigenvalues of Schr\"odinger operators on finite and infinite intervals

TL;DR

This paper investigates the eigenvalues of Schr"odinger operators with integrable potentials on finite and infinite intervals, establishing relationships and estimates that connect eigenvalues to the potentials involved.

Contribution

It introduces new relationships and eigenvalue estimates for Schr"odinger operators with compactly supported potentials on various intervals.

Findings

Derived eigenvalue estimates in terms of potentials

Established relationships between eigenvalues of different operators

Analyzed eigenvalues for operators on finite and infinite intervals

Abstract

We consider a Sturm-Liouville operator a with integrable potential $q$ on the unit interval $I=[0,1]$. We consider a Schr\"odinger operator with a real compactly supported potential on the half line and on the line, where this potential coincides with $q$ on the unit interval and vanishes outside $I$. We determine the relationships between eigenvalues of such operators and obtain estimates of eigenvalues in terms of potentials.