Transition from 3D to 1D in Bose Gases at Zero Temperature
  We investigate the effects of dimensional reduction in Bose gases induced by a strong harmonic confinement in the transverse cylindric radial direction. By using a generalized Lieb-Liniger theory, based on a variational treatment of the transverse width of the Bose gas, we analyze the transition from a 3D Bose-Einstein condensate to the 1D Tonks-Girardeau gas. The sound velocity and the frequency of the lowest compressional mode give a clear signature of the regime involved. We study also the case of negative scattering length deriving the phase diagram of the Bose gas (uniform, single soliton, multi soliton and collapsed) in toroidal confinement. 
L. Salasnich, A. Parola, L. Reatto