Eigenvalue spacing for 1D singular Schr\"odinger operators

TL;DR

This paper develops uniform estimates for eigenvalue spacings of 1D semiclassical Schrödinger operators with singular potentials, simplifying analysis through new semiclassical measures that avoid complex WKB expansions.

Contribution

It introduces a novel semiclassical measure framework that streamlines eigenvalue spacing analysis for singular potentials in 1D Schrödinger operators.

Findings

Uniform eigenvalue spacing estimates established

Simplified analysis without detailed WKB expansions

New semiclassical measures for singular potentials

Abstract

The aim of this paper is to provide uniform estimates for the eigenvalue spacings of one-dimensional semiclassical Schr\"odinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schr\"odinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.