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

We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order \( N^{\kappa -1}\) $N_{}^{\kappa -1}$ , for some \(\kappa >0\) $\kappa >0$ . For the range \( \kappa \in [0, \frac{1}{43})\) $\kappa \in [0,\frac{1}{43})$ , 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 \( \kappa \in [0, \frac{1}{20})\) $\kappa \in [0,\frac{1}{20})$ 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).