Normal and Generalized Bose Condensation in Traps: One Dimensional Examples

Abstract

We prove the following results. (i) One-dimensional Bose gases which interact via unscaled integrable pair interactions and are confined in an external potential increasing faster than quadratically undergo a complete generalized Bose-Einstein condensation (BEC) at any temperature, in the sense that a macroscopic number of particles are distributed on ao(N) number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency ω by ωN/N gives rise to a phase transition ata≡ωβ = 1 in the noninteracting gas. Fora < 1 the limit distribution ofn ₀/N ^a is exponential and 〈n ₀〉/N ^a →  1. Fora > 1 there is BEC with a condensate density 〈n ₀〉/N →  1 − a ^-1. Fora   1, (N/N)(n ₀−〈n ₀〉) is asymptotically distributed following Gumbel's law. For anya > 0 the free energy is −(π²/6βa)N/N+o(N/N), with no singularity ata = 1. (iii) In Model (ii) both above and below the critical temperature the gas undergoes a complete generalized BEC, thus providing a coexistence of ordinary and generalized condensates below the critical point. (iv) Adding an interaction 〈U _N 〉 = o(N N) to Model (ii) we prove that a complete generalized BEC occurs for any β>0.