A Levinson theorem for scattering from a Bose-Einstein condensate

TL;DR

This paper derives a generalized Levinson theorem linking bound excitations in a Bose-Einstein condensate to scattering phase shifts, within the Bogoliubov model, including complex-energy states and continuum bound states.

Contribution

It introduces an exact, generalized Levinson theorem for Bose-Einstein condensates within the Bogoliubov framework, addressing complex and continuum states.

Findings

Derived a relation between bound excitations and phase shifts.

Discussed features like complex-energy and continuum bound states.

Provided a numerical example illustrating the theorem.

Abstract

A relation between the number of bound collective excitations of an atomic Bose-Einstein condensate and the phase shift of elastically scattered atoms is derived. Within the Bogoliubov model of a weakly interacting Bose gas this relation is exact and generalises Levinson's theorem. Specific features of the Bogoliubov model such as complex-energy and continuum bound states are discussed and a numerical example is given.