Zero Energy Ground State in the Three-Body System

TL;DR

This paper investigates the conditions under which a three-particle quantum system in three-dimensional space can have a zero energy ground state, showing that certain resonances prevent such states and exploring how to tune system parameters.

Contribution

It proves that zero energy resonances in any pair of particles prevent the existence of a square integrable zero energy ground state in the three-body system, complementing prior results.

Findings

Zero energy resonances in a pair prevent three-body zero energy ground states.

Coupling constants can be tuned to achieve specific spectral properties.

Existence of negative energy ground states with controlled localization.

Abstract

We consider a 3--body system in $\mathbb{R}^3$ with non--positive potentials and non--negative essential spectrum. Under certain requirements on the fall off of pair potentials it is proved that if at least one pair of particles has a zero energy resonance then a square integrable zero energy ground state of three particles does not exist. This complements the analysis in \cite{1}, where it was demonstrated that square integrable zero energy ground states are possible given that in all two--body subsystems there is no negative energy bound states and no zero energy resonances. As a corollary it is proved that one can tune the coupling constants of pair potentials so that for any given $R, \epsilon >0$: (a) the bottom of the essential spectrum is at zero; (b) there is a negative energy ground state $\psi(\xi)$, where $\int |\psi(\xi)|^2 = 1$; (c) $\int_{|\xi| \leq R} |\psi(\xi)|^2 < \epsilon$.