The Ground State Energy of a Dilute Bose Gas in Dimension n >3

2014/01/23 by Anders Aaen, Aaen, Anders
Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mathematical Physics (math-ph) #Optical properties and cooling technologies in crystalline materials #Quantum, superfluid, helium dynamics

paper · pdf · doi:10.48550/arxiv.1401.5960

openalex publication_date 2014/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a Bose gas in spatial dimension n>3 with a repulsive, radially symmetric two-body potential V. In the limit of low density ρ, the ground state energy per particle in the thermodynamic limit is shown to be (n-2)|\mathbb S^n-1|a^n-2ρ, where |\mathbb S^n-1| denotes the surface measure of the unit sphere in ℝⁿ and a is the scattering length of V. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term (1+Cγ)8π⁴a⁶ρ²|ln(a⁴ρ)| in dimension n=4, where γ:=∫ V(x)|x|^-2 dx, and a correction term which is O(ρ²) in higher dimensions.

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