On spectral estimates for two-dimensional Schrödinger operators

Abstract

For the two-dimensional Schrödinger operator ${\mathrm{H}}_{\alpha V}=-\Delta -\alpha V,V\ge 0$, we study the behavior of the number $N_{-}({\mathrm{H}}_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity. A wide class of potentials is described, for which $N_{-}({\mathrm{H}}_{\alpha V})$ has the semi-classical behavior, i.e. $N_{-}({\mathrm{H}}_{\alpha V})=O(\alpha )$. For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law.