  A quantum field-theoretical model which describes spatially non-homogeneous
one-dimensional non-relativistic repulsive Bose gas in an external harmonic
potential is considered. We calculate the two-point thermal correlation
function of the Bose gas in the framework of the functional integration
approach. The calculations are done in the coordinate representation. A method
of 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 calculated in one loop approximation. The
functional integral representation for the correlation function is obtained in
terms of the low-energy variables, and is estimated by means of the stationary
phase approximation. The asymptotics of the correlation function is studied in
the limit when the temperature is going to zero while the volume occupied by
non-homogeneous Bose gas infinitely increases. It is demonstrated that the
behaviour of the thermal correlation function in the limit described is
power-like, and it is governed by the critical exponent which depends on the
spatial and thermal arguments.
