We study thermodynamic properties of a one-dimensional ideal Bose gas trapped by a steep potential of an exponential type <$E roman {U(q)~=~U sub 0 [e sup {(2q "/" a) sup 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.

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