On the negative spectrum of two-dimensional Schr\"odinger operators with radial potentials

TL;DR

This paper investigates the asymptotic behavior of negative eigenvalues of two-dimensional Schrödinger operators with radial potentials, establishing conditions for semi-classical growth and Weyl law validity as the coupling parameter increases.

Contribution

It provides necessary and sufficient conditions for the semi-classical growth and Weyl asymptotics of negative eigenvalues in radial two-dimensional Schrödinger operators.

Findings

Conditions for semi-classical growth $N_-(H_{\alpha V})=O(\alpha)$

Criteria for Weyl asymptotic law validity

Analysis of eigenvalue behavior as coupling parameter tends to infinity

Abstract

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$ with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling parameter $\alpha$ tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth $N_-(H_{\alpha V})=O(\alpha)$ and for the validity of the Weyl asymptotic law.