Zero Energy Bound States and Resonances in Three--Particle Systems

TL;DR

This paper investigates the spectral properties of three-particle quantum systems in three dimensions, demonstrating conditions under which zero energy bound states occur or are absent, and how to tune interactions to control these states.

Contribution

It establishes the necessity of absence of zero energy resonances for the existence of zero energy bound states and shows how to tune potentials to achieve specific spectral properties.

Findings

Eigenvalues are absorbed at zero energy threshold under certain conditions.

Zero energy resonances in pairs prevent the existence of three-particle ground states.

Coupling constants can be tuned to control the spectrum and ground state localization.

Abstract

We consider a three-particle system in $\mathbb{R}^3$ with non-positive pair-potentials and non-negative essential spectrum. Under certain restrictions on potentials it is proved that the eigenvalues are absorbed at zero energy threshold given that there is no negative energy bound states and zero energy resonances in particle pairs. It is shown that the condition on the absence of zero energy resonances in particle pairs is essential. Namely, we prove 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. It is also 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 $\psi(\xi)$ such that $\int |\psi(\xi)|^2 d^6 \xi= 1$ and $\int_{|\xi| \leq R} |\psi(\xi)|^2 d^6 \xi < \epsilon$.