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

TL;DR

This paper extends the Calogero-Cohn bounds on the number of bound states for spherically symmetric potentials, showing that less restrictive conditions than monotonicity can be used, allowing for oscillating potentials.

Contribution

It generalizes existing bounds by replacing the monotonicity condition with a broader inequality involving a parameter p, modifying the constants accordingly.

Findings

Bounds depend on a parameter p and angular momentum l

Less restrictive conditions allow oscillating potentials

Constants in bounds are adjusted based on p and l

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 $\int^\infty_0 |V(r)|^{1/2}dr$, 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 $p = 1/2$.