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arXiv:2001.00497 · Reviews in Mathematical Physics

The excitation spectrum of the Bose gas in the Gross–Pitaevskii regime

Chiara Boccato

Published Mar 24, 2020• 7 citations
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Abstract

We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.
DOI
10.1142/s0129055x20600065

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We consider interacting Bose gases trapped in a box Λ=[0;1]3Λ= [0;1]^3Λ=[0;1]3 in the Gross-Pitaevskii limit. Assuming the potential to be weak enough, we establish the validity of Bogoliubov's prediction for the ground state energy and the low-energy excitation spectrum. These notes are based on \cite{BBCS3}, a joint work with C. Boccato, C. Brennecke and S. Cenatiempo.
Abstract We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit N→∞N \rightarrow \infty N→∞ N → ∞ .
Bogoliubov theory {N. N. Bogoliubov, Izv. Akad. Nauk Ser. Fiz. 11, 77 (1947) [J. Phys. (USSR) 11, 23 (1947) (in English)]} provides important predictions for the low energy properties of the weakly interacting Bose gas. Recently, Bogoliubov’s predictions were justified rigorously by Boccato et al. [Acta Math. 222(2), 219–335 (2019)] for translation invariant systems in the Gross–Pitaveskii regime, where N bosons in Λ=[0;1]3⊂R3 interact through a potential whose scattering length is of size N−1. In this article, we review recent results from the work of Brennecke et al. [Ann. Henri Poincaré 23, 1583–1658 (2022)], a joint work with Schlein and Schraven, which extends the analysis for translation invariant systems to systems of bosons in R3 that are trapped by an external potential.
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 Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.