On the calculation of the asymptotics of the two-point correlation function of the one-dimensional Bose gas in the trapping potential

The quantum field-theoretical model, which describes spatially non-homogeneous one-dimensional repulsive Bose gas in an external harmonic potential is considered. The two-point correlation function is calculated in the framework of the functional integration. The corresponding functional integrals are estimated by means of the stationary phase approximation. The asymptotical estimates are obtained in the limit when the temperature is going to zero while the volume occupied by the quasi-condensate is increased. The power-law behavior is found for the correlation function in this limit. It is demonstrated that the power-law behavior is governed by the critical exponent dependent on the spatial arguments.