DIMENSIONAL CROSSOVER FOR THE BOSE-EINSTEIN CONDENSATION

We study Bose-Einstein condensation (BEC) for a non-interacting gas in a very anisotropic trap. Therefore, at enough small temperatures some degrees of freedom ''freeze'', thus reducing the ''effective'' dimens ionality of the system. We mainly focus in quasi-bidimensional traps c haracterized by a surface S in the (x, y) plane. We consider two confi ning mechanisms in the z direction: harmonic and rigid wall potentials . There are not forces parallel to S, excepting the rigid wall at the edges. The most relevant results are: (a) The condensate smoothly sets at T similar to O(T-c/log N), where T-c is the tridimensional condens ation temperature. (b) When BEC is present, also the low-lying excited states have a macroscopic occupation this effect is also present in a quasi-onedimensional harmonic ''cigar-shape'' trap. (c) The condensat ion process is sensitive to the shape of S.