Exponential Control of Excitations for Trapped BEC in the Gross-Pitaevskii Regime

TL;DR

This work proves exponential control of excitations in trapped Bose gases within the Gross-Pitaevskii regime, establishing that the number of particles orthogonal to the GP condensate has uniformly bounded exponential moments on low-energy subspaces. The authors achieve this by renormalizing the excitation Hamiltonian through a generalized Bogoliubov transformation with a carefully chosen kernel $oldsymbol{ eta}$, yielding a lower bound tied to the GP energy and a controlled excitation number. They decompose the renormalized Hamiltonian into leading and error parts, prove precise bounds for each part, and demonstrate cancellations that are crucial in the GP scaling. The results yield strong probabilistic statements about the condensate, including exponential large-deviation bounds for the excitation number, and provide a rigorous bridge from mean-field to GP Bogoliubov theory for trapped three-dimensional Bose gases.

Abstract

We consider trapped Bose gases in three dimensions in the Gross-Pitaevskii regime whose low energy states are well known to exhibit Bose-Einstein condensation. That is, the majority of the particles occupies the same condensate state. We prove exponential control of the number of particles orthogonal to the condensate state, generalizing recent results for translation invariant systems.

Theorems & Definitions (21)

• Theorem 1.2
• Remark 1.3: Moment and Exponential Moment Bounds
• Remark 1.4: Moment generating function
• Remark 1.5: Probabilistic interpretation
• Remark 1.6: Large deviations
• Theorem 2.1
• proof
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3