# Bogoliubov theory for dilute bose gases: the Gross-Pitaevskii Regime

## TL;DR

This paper rigorously analyzes Bogoliubov's heuristic theory for dilute Bose gases in the Gross-Pitaevskii regime, confirming its predictions and extending previous results to more general interaction potentials.

## Contribution

It provides a rigorous mathematical validation of Bogoliubov's theory for dilute Bose gases in the Gross-Pitaevskii limit, removing previous restrictions on interaction strength.

## Key findings

- Validation of Bogoliubov's predictions in the Gross-Pitaevskii regime
- Extension of previous results to non-small interaction potentials
- Rigorous control of corrections beyond Bogoliubov theory

## Abstract

In 1947 Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely to test the validity of Bogoliubov's predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in [10] and [11], joint works with C. Boccato, C. Brennecke, and B. Schlein, which extend a previous result obtained in [8], removing the assumption of small interaction potential.