Inequalities on the number of bound states in oscillating potentials

For short-range oscillating potentialsV(r), such that possesses some regularity properties we establish inequalities on the number of bound states. In particular we show that by replacingV(r) by –4(W(r))² in the classical inequalities we get bounds for this new class of potentials. Optimal bounds are also obtained. The behaviour for large coupling constants is studied.