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We show that the transition from a normal conducting state to asuperconducting state is a second-order phase transition in the BCS-Bogoliubovmodel of superconductivity from the viewpoint of operator theory. Here we haveno magnetic field. Moreover we obtain the exact and explicit expression for thegap in the specific heat at constant volume at the transition temperature. Tothis end, we have to differentiate the thermodynamic potential with respect tothe temperature two times. Since there is the solution to the BCS-Bogoliubovgap equation in the form of the thermodynamic potential, we have todifferentiate the solution with respect to the temperature two times.Therefore, we need to show that the solution to the BCS-Bogoliubov gap equationis differentiable with respect to the temperature two times as well as itsexistence and uniqueness. We carry out its proof on the basis of fixed pointtheorems.

An operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity (final version)