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^ -1 N κ - 1, for some >0 κ > 0. For the range [0, 143) 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 [0, 120) 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).