Rigorous conditions for the existence of bound states at the threshold in the two-particle case

TL;DR

This paper rigorously analyzes the conditions under which weakly bound states in two-particle quantum systems approach the continuum threshold without spreading, using Green's functions to establish bounds on wave function behavior.

Contribution

It provides a rigorous proof that for certain potentials, bound states approaching the threshold do not spread and become true threshold bound states, with methods applicable to many-body systems.

Findings

Bound states do not spread for potentials decaying slower than 1/r^2.

Bound states at the threshold are proven to exist under specific decay conditions.

The method is extendable to many-body quantum systems.

Abstract

In the framework of non-relativistic quantum mechanics and with the help of the Greens functions formalism we study the behavior of weakly bound states as they approach the continuum threshold. Through estimating the Green's function for positive potentials we derive rigorously the upper bound on the wave function, which helps to control its falloff. In particular, we prove that for potentials whose repulsive part decays slower than $1/r^{2}$ the bound states approaching the threshold do not spread and eventually become bound states at the threshold. This means that such systems never reach supersizes, which would extend far beyond the effective range of attraction. The method presented here is applicable in the many--body case.