  By exact numerical solutions of the Gross-Pitaevskii (GP) equation in 3D, we
assess the validity of 1D and 2D approximations in the study of Bose-Einstein
condensates confined in harmonic trap potentials. Typically, these
approximations are performed when one or more of the harmonic frequencies are
much greater than the remaining ones, using arguments based on the adiabatic
evolution of the initial approximated state. Deviations from the 3D solution
are evaluated as a function of both the effective interaction strength and the
ratio between the trap frequencies that define the reduced dimension where the
condensate is confined. The observables analyzed are both stationary and
dynamical character, namely, the chemical potential, the wave function
profiles, and the time evolution of the approximated 1D and 2D stationary
states, considered as initial states in the 3D GP equation. Our study, besides
setting quantitative limits on approximations previously developed, should be
useful in actual experimental studies where quasi-1D and quasi-2D conditions
are assumed. From a qualitative perspective, 1D and 2D approximations certainly
become valid when the anisotropy is large, but in addition, the interaction
strength needs to be above a certain threshold.
