Generalization of the Calogero-Cohn Bound on the Number of Bound States

Abstract

It is shown that for the Calogero-Cohn type upper bounds on the number of bound states of a negative spherically symmetric potential $V(r)$, in each angular momentum state, that is, bounds containing only the integral ${\int }_0^{\infty }|V(r){|}^{1/2}dr$, the condition $V^{\prime }(r)\ge 0$ is not necessary, and can be replaced by the less stringent condition $(d/dr)[r^{1-2p}(-V{)}^{1-p}]\le 0,1/2\le p<1$, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on $p$ and $\ell$, and tend to the standard value for $p=1/2$.

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