  We study a gas of bosonic dipolar atoms in the presence of a transverse
harmonic trapping potential by using an improved variational Bethe ansatz,
which includes the transverse width of the atomic cloud as a variational
parameter. Our calculations show that the system behavior evolves from
quasi-one dimensional to a strictly one-dimensional one by changing the
atom-atom interaction, or the axial density, or the frequency of the transverse
confinement. Quite remarkably, in the droplet phase induced by the attractive
dipolar interaction the system becomes sub-one dimensional when the transverse
width is smaller than the characteristic length of the transverse harmonic
confinement.
