Bose–Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross–Pitaevskii Regime

Abstract

We consider a Bose gas consisting of N particles in R 3 , trappedbyanexternalf ieldandinteractingthroughatwo−

Bogoliubov theory in the Gross-Pitaevskii limit: a simplified approach

“…Recently, a rigorous version of Bogoliubov theory [8] has been developed in [4, 5, 6, 7] to provide more precise information about the low-energy spectrum of in equation (1), resolving the ground state energy and low-lying excitations up to errors that vanish in the limit ; and about the corresponding eigenvectors, showing Bose-Einstein condensation with optimal control over the number of orthogonal excitations. Analogous results have also been established for Bose gases trapped by external potentials in the Gross-Pitaevskii regime [10, 11, 23, 25] and for Bose gases in scaling limits interpolating between the Gross-Pitaevskii regime and the thermodynamic limit [1, 9]. Very recently, the upper bound for the ground state energy has also been extended to the case of hard-sphere interaction, as announced in [2].…”

The Bose gas in a box with Neumann boundary conditions

“…The Gross-Pitaevskii regime has been studied for the translation invariant Bose gas, where periodic boundary conditions are imposed on Λ 1 , and for the trapped Bose gas, where particles move in R 3 and are confined by an external potential. In these cases, Bose-Einstein condensation has been proved [23,24,28] with optimal rate [5,7,27,11,18]. In the translation invariant case, the ground state energy has been shown in [6] to be is a boundary contribution.…”

The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime

“…i) To keep our analysis as simple as possible, we restrict out attention to bosons moving in the two-dimensional unit torus. Our results could be extended to more general trapping potentials, combining the proof of Theorem 1.1 with ideas from [30,13,31,14], recently developed in the three dimensional setting.…”