Universal bound states and resonances with Coulomb plus short-range potentials

TL;DR

This paper develops a zero-range theory for charged particles with Coulomb and short-range interactions, revealing universal bound states and resonances, including infinite resonances for repulsive Coulomb and infinite bound states for attractive Coulomb potentials.

Contribution

It generalizes the Bethe-Peierls boundary condition to Coulomb plus short-range potentials, providing a universal framework for low-energy charged particle interactions.

Findings

Infinite resonances for repulsive Coulomb potential.

Infinite bound states for attractive Coulomb potential.

Inverse-square attraction induces deep bound states in three-particle systems.

Abstract

We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.