Breathing mode collective excitation of a Bose-Einstein condensate in a low-dimensional trap

Using a sum-rule method and a time-dependent variational method, we study the breathing mode collective-excitation frequency of a trapped Bose-Einstein condensate. We show that the general result for a three-dimensional trap correctly goes over to purely two- or one-dimensional limit. In the case of a realistic two- or one-dimensional trap, we obtain the lowest order correction for the collective-excitation frequency due to the finiteness of the trap frequency in the tightly trapped direction. We also show numerical results for the collective-excitation frequencies using parameters relevant to recent experiments