Smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity

Abstract

We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs to the closed interval $[0,\tau ]$ with $\tau >0$ nearly equal to half of the transition temperature. We show that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotone decreasing with respect to the temperature. Moreover, we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy, or that the solution is approximated by such a smooth function.

arXiv comment15 pages. arXiv admin note: substantial text overlap with arXiv:1607.00090, arXiv:1411.7473

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• Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity
• An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature
• The BCS-Bogoliubov gap equation with external magnetic field and the first-order phase transition