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On spectral estimates for two-dimensional Schrödinger operators

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