The BCS Functional for General Pair Interactions

Abstract

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. 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 linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.

Revised Version. To appear in Commun. Math. Phys.

References in corpus (4)

• BCS-BEC Crossover: From High Temperature Superconductors to Ultracold Superfluids
• Generalized Hartree-Fock Theory and the Hubbard Model
• Density-induced BCS to Bose-Einstein crossover
• BCS-BEC crossover with a finite-range interaction

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