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arXiv · 2401.00784

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

Christian Brennecke·Morris Brooks·Cristina Caraci·Jakob Oldenburg
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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 $ N^{\kappa-1}$, for some $\kappa >0$. For the range $ \kappa\in [0, \frac1{43})$, \cite{ABS} 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, \frac1{20})$ by combining some of the key ideas of \cite{ABS} with the novel diagonalization approach introduced recently in \cite{Br}, which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than \cite{ABS}.

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Christian Brennecke, Morris Brooks, Cristina Caraci, Jakob Oldenburg. 2024-01-01. A Short Proof of Bose-Einstein Condensation in the Gross-Pitaevskii Regime and Beyond. https://doi.org/10.1007/s00023-024-01465-8

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