Sufficient conditions for the existence of bound states in a central potential

TL;DR

This paper develops new, more stringent sufficient conditions for the existence of bound states in non-positive central potentials, providing tighter bounds on the critical coupling strength needed for bound state formation.

Contribution

It introduces a broad class of sufficient conditions that improve upon previous bounds for the critical coupling constant in central potentials.

Findings

Derived upper limits on the critical coupling constant g_c^{(l)}

Conditions applicable to a wide class of non-positive central potentials

Results are more stringent than previously known bounds

Abstract

We show how a large class of sufficient conditions for the existence of bound states, in non-positive central potentials, can be constructed. These sufficient conditions yield upper 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. These upper limits are significantly more stringent than hitherto known results.