  We show that the transition from a normal conducting state to a
superconducting state is a second-order phase transition in the BCS-Bogoliubov
model of superconductivity from the viewpoint of operator theory. Here we have
no magnetic field. Moreover we obtain the exact and explicit expression for the
gap in the specific heat at constant volume at the transition temperature. To
this end, we have to differentiate the thermodynamic potential with respect to
the temperature two times. Since there is the solution to the BCS-Bogoliubov
gap equation in the form of the thermodynamic potential, we have to
differentiate the solution with respect to the temperature two times.
Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation
is differentiable with respect to the temperature two times as well as its
existence and uniqueness. We carry out its proof on the basis of fixed point
theorems.
