Upper bounds on eigenvalue spacing for decaying potentials
Milivoje Lukic, Brian Simanek
TL;DR
This work bounds the local Dirichlet eigenvalue spacing of half-line Schrödinger operators $H=-\frac{d^2}{dx^2}+V$ in terms of the decay rate of the potential via the growth of $\int_0^x |V(t)|\,dt$. Using Prüfer variables, the authors show that a decaying potential imposes lower bounds on the separation of eigenvalues of $H_X$ on finite intervals, quantified by a function $h(x)=O\left(\frac{1}{x}\left(1+\int_0^x |V(t)|\,dt\right)\right)$. The main result asserts that for any $[\alpha,\beta]\subset[a,\infty)$ with $|\beta-\alpha|\ge h(X)$, there is at least one Dirichlet eigenvalue of $H_X$ in $[\alpha^2/4,\beta^2/4]$, linking spectral regularity to potential decay. The paper also situates this bound in the broader context of bulk universality and singular spectral measures, highlighting how weaker decay leads to more clustered eigenvalues and potential spectral transitions.
Abstract
We study decaying half-line Schrödinger 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.