The Low Energy Spectrum of Trapped Bosons in the Gross-Pitaevskii Regime

TL;DR

This paper reviews recent rigorous results confirming Bogoliubov theory's predictions for the low energy spectrum of trapped weakly interacting Bose gases in the Gross-Pitaevskii regime, extending previous translation-invariant analyses.

Contribution

It extends the rigorous analysis of Bogoliubov theory to trapped Bose gases in 3, building on prior work for translation-invariant systems.

Findings

Confirmed Bogoliubov predictions for trapped systems

Extended analysis from translation-invariant to trapped gases

Provided mathematical rigor for low energy spectrum

Abstract

Bogoliubov Theory provides important predictions for the low energy properties of the weakly interacting Bose gas. Recently, Bogoliubov's predictions could be justified rigorously in \cite{BBCS3} for translation invariant systems in the Gross-Pitaveskii regime, where $N$ bosons in $\Lambda = [0;1]^3\subset \mathbb{R}^3$ interact through a potential whose scattering length is of size $ N^{-1}$. In this note, we review recent results from \cite{BSS2}, a joint work with B. Schlein and S. Schraven, which extends the analysis of \cite{BBCS3} to systems of bosons in $\mathbb{R}^3$ that are trapped by an external potential.