Lower bounds on the ground-state energy and necessary conditions for the existence of bound states: The potential problem

We consider a particle in a three-dimensional potential V(r) or in a one-dimensional potential V(x). We use a Green's-function approach to show that for many V's of interest one can obtain a necessary condition for the existence of a bound state below a specified energy that is somewhat stronger than the usual condition; furthermore, results can be obtained for V(r){≠}V(r), under some special restrictions, which utilize an integral containing only one, not two, Green's functions.