The Bose-Einstein condensation is investigated for N particles in a pa
rabolic confining potential. The partition function in the grand-canon
ical as well as in the canonical ensemble is obtained analytically, an
d the thermodynamical quantities are studied for the one-dimensional (
1D) and for the three-dimensional (3D) version of the model. In 1D, a
dramatic rise in the ground state occupancy with decreasing temperatur
e is predicted. Both the grand-canonical and the canonical ensemble yi
eld the same behavior of the specific heat as a function of the temper
ature and the number of particles. There is no indication of a critica
l phenomenon for a finite number of particles, neither in the ground s
tate occupancy nor in the specific heat. Also in 3D, both ensembles gi
ve an identical contribution for the specific heat, but showing thereb
y a phase transition with a critical temperature which scales with the
cube root of the number of particles, as predicted before by semi-cla
ssical continuum models. Copyright (C) 1996 Elsevier Science Ltd