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

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.

References in corpus (1)

• Evidence for Efimov quantum states in an ultracold gas of cesium atoms

Cited by in corpus (5)

• Bound States at Threshold resulting from Coulomb Repulsion
• Zero Energy Bound States and Resonances in Three--Particle Systems
• Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality
• Universal Angular Probability Distribution of Three Particles near Zero Energy Threshold
• Quantum Systems at The Brink