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

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

1995-11-02

arxiv.org/abs/hep-th/9511012v2

Physics

High Energy Physics

High Energy Physics - Theory

1 page. Correctly formatted version (replaces previous version)

Scientific paper

10.1063/1.531450

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 $\int^\infty_0 |V(r)|^{1/2}dr$, the condition $V'(r) \geq 0$ is not necessary, and can be replaced by the less stringent condition $(d/dr)[r^{1-2p}(-V)^{1-p}] \leq 0, 1/2 \leq p < 1$, 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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