### Abstract

In this paper, we present a mathematical analysis for the phonon-dominated multiband isotropic and anisotropic BCS gap equations at any finite temperature T. We establish the existence of a critical temperature T_{c} so that, when T < T_{c}, there exists a unique positive gap solution, representing the superconducting phase; when T > T_{c}, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T = T_{c}, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T < T_{c} monotonically and continuously. In particular, as T → T_{c}, the gap solution tends to zero, which enables us to determine the critical temperature T_{c}. In the isotropic case where the entries of the interaction matrix K are all constants, we are able to derive an elegant T_{c} equation which says that T_{c} depends only on the largest positive eigenvalue of K but does not depend on the other details of K. In the anisotropic case, we may derive a similar T_{c} equation in the context of the Markowitz-Kadanoff model and we prove that the presence of anisotropic fluctuations enhances T_{c} as in the single-band case. A special consequence of these results is that the half-unity exponent isotope effect may rigorously be proved in the multiband BCS theory, isotropic or anisotropic.

Original language | English (US) |
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Pages (from-to) | 60-74 |

Number of pages | 15 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 200 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2005 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics