Ideal Bose gas in steep one-dimensional traps

We study thermodynamic properties of a one-dimensional ideal Bose gas trapped by a steep potential of an exponential type U(q) = U-0 [e((2q/a)b)-1 ]. Fugacity, energy, and heat capacity of such a system are calculated for various combinations of the potential parameters as well for several values of the number of particles N. Both the thermodynamic limit and finite N are considered. Estimations for the single-particle spectrum asymptotics are obtained in the analytical form involving the Lambert W function. In the thermodynamic limit, the Bose-Einstein condensation is predicted for 0 < b < 2. We associate such behavior with an effective temperature-dependent space dimensionality arising due to the influence of the external potential of the analyzed type.