Exact quantum partition function of the BCS model

The exact calculation of the reduced BCS model quantum partition function (QPF) in the thermodynamic limit is carried out by the path integration method. The expression for the QPF and the phase transition temperatureT_c in the regular phase coincide with the results of Bogolyubov. In the nonregular phase a temperature singularity appears in the expression for the QPF: the QPF diverges in the region of temperaturesT_c which are smaller than some critical temperatureT_c^*, and it turns out that in all casesT_c^* >T_c and the differenceT_c^*–T_c is not small. The interpretation of the temperatureT_c^* is given.