Semiclassical asymptotics for a class of singular Schrödinger operators

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrödinger 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.

Dedicated to Ari Laptev on the occasion of his 70th birthday