The excitation spectrum for weakly interacting bosons in a trap

TL;DR

This paper rigorously analyzes the low-energy excitation spectrum of a trapped weakly interacting Bose gas, confirming the Bogoliubov approximation's predictions by relating the spectrum to an effective one-particle operator.

Contribution

It provides a rigorous proof that the excitation spectrum matches the eigenvalues of an effective operator, validating the Bogoliubov approximation for trapped Bose gases.

Findings

Spectrum described by eigenvalues of an effective operator

Validation of Bogoliubov approximation in a trap

Rigorous mathematical proof of spectral properties

Abstract

We investigate the low-energy excitation spectrum of a Bose gas confined in a trap, with weak long-range repulsive interactions. In particular, we prove that the spectrum can be described in terms of the eigenvalues of an effective one-particle operator, as predicted by the Bogoliubov approximation.