Critical strength of attractive central potentials

TL;DR

This paper derives sequences of necessary and sufficient conditions to determine the critical coupling strength at which bound states first appear in attractive central potentials, providing bounds that converge to the exact critical value.

Contribution

It introduces new sequences of bounds on the critical coupling for bound state formation in attractive central potentials, improving understanding of the threshold conditions.

Findings

Sequences of bounds on critical coupling converge to the exact value.

Provides necessary and sufficient conditions for bound state existence.

Applicable to various attractive central potentials.

Abstract

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yields several sequences of upper and lower limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears, which converges to the exact critical value.