Necessary and sufficient conditions for existence of bound states in a central potential

TL;DR

This paper derives necessary and sufficient conditions for the existence of bound states in arbitrary central potentials using the Birman-Schwinger method, providing bounds on the critical potential strength for bound state formation.

Contribution

It introduces a series of necessary conditions and a sufficient condition for bound states in central potentials, with converging bounds on the critical potential strength.

Findings

Series of lower bounds on critical potential strength converging to the exact value

Upper bound on critical strength for monotonic potentials

Conditions applicable to arbitrary central potentials

Abstract

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.