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

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 $\kappa \in [0,\frac{1}{43})$, \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 $\kappa \in [0,\frac{1}{20})$ 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}.

13 pages + one appendix; accepted journal version

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Cited by in corpus (2)

• Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime
• Derivation of the Gross-Pitaevskii Dynamics through Renormalized Excitation Number Operators