# Mathematics Colloquia and Seminars

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### The B-C-S model for general pair interactions and its critical temperature

**Mathematical Physics & Probability**

Speaker: | Christian Hainzl, University of Alabama at Birmingham |

Location: | 2112 MSB |

Start time: | Tue, May 15 2007, 2:10PM |

We present a rigorous analysis of the Bardeen-Cooper-Schrieffer (BCS) model for general pair interaction potentials \lambda V. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain effective linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. We give necessary and sufficient conditions for the existence of a positive critical temperature. Moreover we present a precise calculation of the asymtotic behavior of the critical temperature in the limit \lambda to 0, which is {\em not} governed by the scattering length.