arXiv:2501.15803 · Letters in Mathematical Physics

Exponential control of excitations for trapped BEC in the Gross–Pitaevskii regime

Nils Behrmann, Christian Brennecke, Simone Rademacher

Published Aug 21, 2025• 1 citation

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 from Nam and Rademacher (Trans Am Math Soc, 2023, arXiv:2307.10622) for translation invariant systems.

DOI: 10.1007/s11005-025-01986-1

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arXiv:2501.15803 · Letters in Mathematical Physics

Exponential control of excitations for trapped BEC in the Gross–Pitaevskii regime

Nils Behrmann, Christian Brennecke, Simone Rademacher

Published Aug 21, 2025• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1007/s11005-025-01986-1

Related papers

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t\mapsto e^{-B_t} e^{-iH_Nt}

with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics

t\mapsto e^{-iH_Nt}

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}

N κ - 1 , for some

\kappa >0

κ > 0 . For the range

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

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

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

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).