  Recent experimental and theoretical work has indicated conditions in which a
trapped, low-density Bose gas ought to behave like the 1D delta-function Bose
gas solved by Lieb and Liniger. Up to now the theoretical arguments have been
based on variational - perturbative ideas or numerical investigations. There
are 4 parameters: density, transverse and longitudinal dimensions, and
scattering length. In this paper we explicate 5 parameter regions in which
various types of 1D or 3D behavior occur. Our discussion is based on a rigorous
analysis of the many-body Schroedinger equation.
  This version of the article is a combination of the earlier 4-page arXiv
version and the revised 4-page Physical Review Letters version.
