On boson condensation considering a generalized Casimir example

Abstract

From the concept of generalized condensation [M. Van den Berg, J. T. Lewis, and J. V. Pulè, Helv. Phys. Acta 59, 1271 (1986)] it is known that two critical densities ρc and ρm exist for a free boson gas. Density ρc is the classical one and ρm is the critical density below which there can be no macroscopic occupation of ground state. A free boson gas is studied in a weak external potential which behaves asymptotically like ‖x1‖α1+‖x2‖α2+⋅⋅⋅ +‖xd‖αd near the origin. It is shown that there are only two possibilities to get ρc<ρm<∞, namely, α1=α2=∞ and d≥3 (this corresponds to Dirichlet boundary conditions), and α1=2 and d≥2 (i.e., a harmonic oscillator).