A mathematical proof that the transition to a superconducting state is a second-order phase transition

We deal with the gap function and the thermodynamical potential in the BCSBogoliubov theory of superconductivity, where the gap function is a function of the temperature T only. We show that the squared gap function is of class C on the closed interval [ 0, Tc ] and point out some more properties of the gap function. Here, Tc stands for the transition temperature. On the basis of this study we then give, examining the thermodynamical potential, a mathematical proof that the transition to a superconducting state is a second-order phase transition. Furthermore, we obtain a new and more precise form of the gap in the specific heat at constant volume from a mathematical point of view. Mathematics Subject Classification (2000): 45G10, 82D55