Interaction-induced crossover versus finite-size condensation in a weakly interacting trapped one-dimensional Bose gas

We discuss the transition from a fully decoherent to a (quasi)condensate regime in a harmonically trapped weakly interacting one-dimensional (1D) Bose gas. By using analytic approaches and verifying them against exact numerical solutions, we find a characteristic crossover temperature and crossover atom number that depend on the interaction strength and the trap frequency. We then identify the conditions for observing either an interaction-induced crossover scenario or else a finite-size Bose-Einstein condensation phenomenon characteristic of an ideal trapped 1D gas.