Semiclassical asymptotics for a class of singular Schr\"odinger operators

TL;DR

This paper derives a two-term semiclassical asymptotic formula for the eigenvalues of Schrödinger operators with singular potentials near the boundary of a bounded domain, extending understanding of spectral properties in singular settings.

Contribution

It introduces a novel asymptotic analysis for Schrödinger operators with boundary-singular potentials, providing explicit eigenvalue sum formulas under weak assumptions.

Findings

Two-term asymptotic eigenvalue formula derived

Applicable to operators with potentials behaving like the inverse square of distance to boundary

Extends spectral analysis to singular Schrödinger operators

Abstract

Let $\Omega \subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schr\"odinger operators $-\Delta+ W$ on $\Omega$ with $W(x)\approx\mathrm{dist}(x, \partial\Omega)^{-2}$ as $\mathrm{dist}(x, \partial\Omega)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.