Upper bounds on eigenvalue spacing for decaying potentials

TL;DR

This paper investigates how the decay rate of potentials in Schrödinger operators influences the upper bounds of eigenvalue spacing on finite intervals, linking spectral decay to eigenvalue distribution.

Contribution

It establishes a connection between potential decay rates and eigenvalue spacing bounds for Schrödinger operators with singular spectra.

Findings

Decay rate determines upper bounds on eigenvalue spacing.

Singular spectral measures exhibit varied local eigenvalue behaviors.

Potential decay influences spectral properties and eigenvalue distribution.

Abstract

We study decaying half-line Schr\"odinger operators and the local eigenvalue spacing of their Dirichlet restrictions. While absolutely continuous spectrum is strongly associated with bulk universality and clock behavior, singular spectral measures can correspond to varied local behaviors. In this work, the rate of decay of the potential is shown to give upper bounds for the spacing of Dirichlet eigenvalues on finite intervals.