A sufficient condition for the existence of bound states for scalar spherically symmetric potentials

Abstract

We consider the class of three-dimensional attractive finite range or similar potentials , depending on a strength constant . Beyond a `critical' value, , the potential has at least one bound state. For the s-wave, we propose simple bounds for obtained by formally solving the Schrodinger equation for the zero energy bound state. The various bounds are compared with the exact value for a set of usual potentials. They are also compared with bounds derived in earlier works by Glaser et al and by Calogero. We show that the case is solved by equations very similar to ones.