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arXiv:2307.11062 · Journal of Mathematical Physics

Exponential decay of the number of excitations in the weakly interacting Bose gas

David Mitrouskas, Peter Pickl

Published Dec 1, 2023• 3 citations
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Abstract

We consider N trapped bosons in the mean-field limit with coupling constant λN = 1/(N − 1). The ground state of such systems exhibits Bose–Einstein condensation. We prove that the probability of finding ℓ particles outside the condensate wave function decays exponentially in ℓ.
DOI
10.1063/5.0172199

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We study the mean-field dynamics of a system of NNN interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having nnn particles outside the condensate decays exponentially in nnn for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having nnn excitations.
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We consider a trapped dilute gas of NNN bosons in R3\mathbb{R}^3R3 interacting via a three-body interaction potential of the form N V(N1/2(x−y,x−z))N\, V(N^{1/2}(x-y,x-z))NV(N1/2(x−y,x−z)). In the limit N→∞N\to \inftyN→∞, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schrödinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the NNN-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schrödinger minimizer is unique up to a complex phase.
The properties and stability of a trapped Bose-Einstein condensate are strongly influenced by attractive interactions between the particles. We describe the spatial distribution, stability, and collisional loss rates for a weakly interacting gas in the mean-field limit. We show how the condensate contracts and becomes unstable as the number of condensate atoms increases. We further show how the number of atoms is limited by the collisional loss rates associated with the contraction of the condensate; this loss is in addition to the particle ejection decay indentified by Kagan et al. \textcopyright{} 1996 The American Physical Society.