On spectral estimates for two-dimensional Schr\"odinger operators

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