The solution to the BCS gap equation and the second-order phase transition in superconductivity

Abstract

The existence and the uniqueness of the solution to the BCS gap equation of superconductivity is established in previous papers, but the temperature dependence of the solution is not discussed. In this paper, in order to show how the solution varies with the temperature, we first give another proof of the existence and the uniqueness of the solution and point out that the unique solution belongs to a certain set. Here this set depends on the temperature $T$. We define another certain subset of a Banach space consisting of continuous functions of both $T$ and $x$. Here, $x$ stands for the kinetic energy of an electron minus the chemical potential. Let the solution be approximated by an element of the subset of the Banach space above. We second show, under this approximation, that the transition to a superconducting state is a second-order phase transition.

Journal of Mathematical Analysis and Applications, in press

References in corpus (5)

• Critical Temperature and Energy Gap for the BCS Equation
• The BCS Functional for General Pair Interactions
• Is the solution to the BCS gap equation continuous in the temperature ?
• A mathematical proof that the transition to a superconducting state is a second-order phase transition
• Superconductivity and the BCS-Bogoliubov Theory

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• Smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity
• An operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity (final version)
• The solution to the BCS gap equation for superconductivity and its temperature dependence
• An operator-theoretical study of the specific heat and the critical magnetic field in the BCS-Bogoliubov model of superconductivity
• Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity
• Lipschitz continuity and monotone decreasingness of the solution to the BCS gap equation for 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