  A quantum field-theoretical model, which describes spatially non-homogeneous
repulsive Bose gas in an external harmonic potential is considered. Two-point
thermal correlation functions of the Bose gas are calculated in the framework
of the functional integration approach. Successive integration over the
``high-energy'' functional variables first and then over the ``low-energy''
ones is used. The effective action functional for the low-energy variables is
obtained in one loop approximation. The functional integral representations for
the correlation functions are estimated by means of the stationary phase
approximation. A power-law asymptotical behaviour of the correlators of the
one-dimensional Bose gas is demonstrated in the limit, when the temperature is
going to zero, while the volume occupied by the non-homogeneous Bose gas
infinitely increases. The power-law behaviour is governed by the critical
exponent dependent on the spatial arguments.
