  A generalization of the semiclassical approximation is suggested allowing for
an essential extension of its region of applicability. In particular, it
becomes possible to describe Bose-Einstein condensation of a trapped gas in
low-dimensional traps and in traps of low confining dimensions, for which the
standard semiclassical approximation is not applicable. The results of the
modified approach are shown to coincide with purely quantum-mechanical
calculations for harmonic traps, including the one-dimensional harmonic trap.
The advantage of the semiclassical approximation is in its simplicity and
generality. Power-law potentials of arbitrary powers are considered. Effective
thermodynamic limit is defined for any confining dimension. The behaviour of
the specific heat, isothermal compressibility, and density fluctuations is
analyzed, with an emphasis on low confining dimensions, where the usual
semiclassical method fails. The peculiarities of the thermodynamic
characteristics in the effective thermodynamic limit are discussed.
