, Volume 31, Issue 3, pp 623-637

On condensation in the free-boson gas and the spectrum of the Laplacian

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We prove the convergence of the thermodynamic functions of a free boson gas for ad-dimensional (d=3,4,...) van Hove sequence of convex regions. The thermodynamic functions behave singularly at a critical densityρ c which is independent of the geometrical details of the sequence. We are led to define a second critical densityρ m depending on the geometrical details of the sequence. For densities betweenρ c andρ m none of the single particle states is macroscopically occupied. We derive a sufficient condition on the sequence such thatρ m c .