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arXiv:2310.05448 · SIAM Journal on Mathematical Analysis

Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross–Pitaevskii Regime

Christian Brennecke, Jinyeop Lee, Phan Thành Nam

Published Jul 19, 2024• 3 citations
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Abstract

Abstract. We consider a translation-invariant system of [Formula: see text] bosons in [Formula: see text] that interact through a repulsive two-body potential with scattering length of order [Formula: see text] in the limit [Formula: see text]. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov’s prediction on the fluctuations around the condensate.

DOI
10.1137/23m1608215

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Abstract. We consider a translation-invariant system of [Formula: see text] bosons in [Formula: see text] that interact through a repulsive two-body potential with scattering length of order [Formula: see text] in the limit [Formula: see text]. We derive second order expressions for the one- and two-particle reduced density matrix matrices of the Gibbs state at fixed positive temperatures, thus obtaining a justification of Bogoliubov’s prediction on the fluctuations around the condensate.
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We consider systems of NNN bosons in R3\mathbb{R}^3R3, trapped by an external potential. The interaction is repulsive and has a scattering length of the order N−1N^{-1}N−1 (Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum up to errors that vanish in the limit N→∞N\to \inftyN→∞.
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