The number of bound states of singular oscillating potentials

Abstract

For a spherically symmetric potential such that rV∈L ¹(a, ∞), ∀a>0, and is such that, if we define W=−∫ ^∞ _r V(t) d(t), W belongs to L ¹ (0, ∞) and rW→0 as r→0, we show that the number of bound states in any partial-wave satisfies the bound n⩽2 ∫ ^∞₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±∣g∣∫ ^∞₀ ∣W(r) dr as g→±∞.