Normal and generalized Bose condensation in traps: One dimensional examples

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 a o(N)number of one-particle states. (ii) In a one dimensional harmonic trap the replacement of the oscillator frequency \omega by \omega\ln N/N gives rise to a phase transition at a=\hbar\omega\beta=1 in the noninteracting gas. For a<1 the limit distribution of n_0/N^a is exponential and <n_0>/N^a tends to 1. For a>1 there is BEC with a condensate density <n_0>/N going to 1-1/a. For a>=1, (\ln N/N)(n_0-<n_0>) is asymptotically distributed following Gumbel's law. For any a>0 the free energy is -(\pi^2/6a\beta)N/\ln N+o(N/\ln N), with no singularity at a=1. (iii) In Model (ii) both above and below the critical temperature the 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\ln N) to Model (ii) we prove that a complete generalized BEC occurs at all temperatures.