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For the two-dimensional Schrödinger operator H˛V D ˛V; V 0; we study the behavior of the number N .H˛V / of its negative eigenvalues (bound states), as the coupling parameter˛tends to infinity. A wide class of potentials is described, for which N .H˛V / has the semi-classical behavior, i.e. N .H˛V / D O.˛/. For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law. Mathematics Subject Classification (2010). Primary 35J10;doi:10.4171/jst/53 fatcat:3txcepvubjcvbgglbswyx4owla