  We first show some properties such as smoothness and monotone decreasingness
of the solution to the BCS-Bogoliubov gap equation for superconductivity.
Moreover we give the behavior of the solution with respect to the temperature
near the transition temperature. On the basis of these results, dealing with
the thermodynamic potential, we then 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 and we need to introduce a cutoff
$\varepsilon>0$, which is sufficiently small and fixed (see Remark
\ref{rmk:varepsilon}). Moreover we obtain the exact and explicit expression for
the gap in the specific heat at constant volume at the transition temperature.
