  We study the Bose-Einstein condensate (BEC) in a 3-D toroidal Mexican hat
trap. By changing the parameters of the potential, or the number of bosons, it
is possible to modify strongly the density profile of the BEC. We consider the
ground state properties for positive and negative scattering length and
calculate the spectrum elementary excitations. We also discuss the macroscopic
phase coherence and superfluidity of the BEC by analyzing vortex states and
their stability.
