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arXiv:2401.00784 · Annales Henri Poincaré

A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond

Christian Brennecke, Morris Brooks, Cristina Caraci, Jakob Oldenburg

Published Jul 1, 2024• 5 citations
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Abstract

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

DOI
10.1007/s00023-024-01465-8

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Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order Nκ−1 N^{\kappa -1}Nκ−1 N κ - 1 , for some κ>0\kappa >0κ>0 κ > 0 . For the range κ∈[0,143) \kappa \in [0, \frac{1}{43})κ∈[0,431​) κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for κ∈[0,120) \kappa \in [0, \frac{1}{20})κ∈[0,201​) κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).
Abstract We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order N−1+κN^{-1+\kappa }N−1+κ N - 1 + κ , for κ>0\kappa >0κ>0 κ > 0 . Assuming that κ∈(0;1/43)\kappa \in (0;1/43)κ∈(0;1/43) κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.
We present a novel approach to the Bogoliubov theory of dilute Bose gases, allowing for an elementary derivation of the celebrated Lee-Huang-Yang formula in the Gross-Pitaevskii regime. Furthermore, we identify the low lying excitation spectrum beyond the Gross-Pitaevskii scaling, extending a recent result (Brennecke et al. in Rev Math Phys 34, 2022) to significantly more singular scaling regimes. Finally, we provide an upper bound on the ground state energy in the Gross-Pitaevskii regime that captures the correct expected order of magnitude beyond the Lee-Huang-Yang formula.
The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number NNN goes to infinity with NaNaNa fixed, where aaa is the scattering length of the interparticle interaction, measured in units of the trap size.