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

Abstract

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

13 pages