2005/10/01 by K. Schnee, Schnee, K., Jakob Yngvason +2
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum, superfluid, helium dynamics #Statistical Mechanics (cond-mat.stat-mech) #Strong Light-Matter Interactions #cond-mat.stat-mech #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0510006
Corrected version. To be published in Communications in Mathematical Physics
openalex publication_date 2005/10/01 · arxiv created 2006/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, N≫ 1, the two-dimensional extension, L, of the gas cloud in the trap, the thickness, h≪ L of the trap, and the scattering length a of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premisses that the confining energy, ∼ 1/h2, is much larger than the internal energy per particle, and a/h→ 0, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length a\rm 2D= hexp(-(\hbox\rm const.)h/a). In the parameter region where a/h≪ |ln(ρh2)|-1, with ρ∼ N/ L2 the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter ∼ Na/h. If |ln(ρh2)|-1\lesssim a/h the coupling parameter is ∼ N |ln(ρh2)|-1 and thus independent of a. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.