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arXiv:2607.00859 · Reviews in Mathematical Physics

On the spherical symmetry and finite-range assumptions of the interaction potential in the low energy study of dilute Bose gases

Lukas Reichmann, Arnaud Triay

Published Oct 26, 2026• 0 citations
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Abstract

We consider a Bose gas on the three-dimensional torus in the Gross–Pitaevskii regime and explain how to remove the assumptions of radiality and compact support on the interaction potential in the proof of Bose–Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction. In particular, we sketch the proof of [55].

DOI
10.1142/s0129055x26610040

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We consider a Bose gas on the three-dimensional torus in the Gross–Pitaevskii regime and explain how to remove the assumptions of radiality and compact support on the interaction potential in the proof of Bose–Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction. In particular, we sketch the proof of [55].
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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 → ∞ .
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.