Spectral Singularities and Zero Energy Bound States

TL;DR

This paper re-analyzes zero-energy scattering phenomena, highlighting spectral singularities and zero-energy bound states, with implications for ultra-cold atoms, nano-structures, and nuclear physics experiments.

Contribution

It introduces the concept of spectral singularities and zero-energy bound states, clarifying their properties and potential observability in controlled scattering experiments.

Findings

Zero-energy bound states have infinite scattering length.

Spectral singularities behave like exceptional points without poles.

Results are applicable to systems with tunable interactions.

Abstract

Single particle scattering around zero energy is re-analysed in view of recent experiments with ultra-cold atoms, nano-structures and nuclei far from the stability valley. For non-zero orbital angular momentum the low energy scattering cross section exhibits dramatic changes depending on the occurrence of either a near resonance or a bound state or the situation in between, that is a bound state at zero energy. Such state is singular in that it has an infinite scattering length, behaves for the eigenvalues but not for the eigenfunctions as an exceptional point and has no pole in the scattering function. These results should be observable whenever the interaction or scattering length can be controlled.