  We investigate the emergence of three-dimensional behavior in a
reduced-dimension Bose-Einstein condensate trapped by a highly anisotropic
potential. We handle the problem analytically by performing a perturbative
Schmidt decomposition of the condensate wave function between the tightly
confined (transverse) direction(s) and the loosely confined (longitudinal)
direction(s). The perturbation theory is valid when the nonlinear scattering
energy is small compared to the transverse energy scales. Our approach provides
a straightforward way, first, to derive corrections to the transverse and
longitudinal wave functions of the reduced-dimension approximation and, second,
to calculate the amount of entanglement that arises between the transverse and
longitudinal spatial directions. Numerical integration of the three-dimensional
Gross-Pitaevskii equation for different cigar-shaped potentials and
experimentally accessible parameters reveals good agreement with our analytical
model even for relatively high nonlinearities. In particular, we show that even
for such stronger nonlinearities the entanglement remains remarkably small,
which allows the condensate to be well described by a product wave function
that corresponds to a single Schmidt term.
