On the negative spectrum of two-dimensional Schrödinger operators with radial potentials

Abstract

For a two-dimensional Schrödinger 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.

13 pages