Critical temperature and energy gap for the BCS equation
Abstract
We derive upper and lower bounds on the critical temperature T_{c} and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin (1)/(2) fermions interacting via a local twobody interaction potential λV(x) . At weak coupling λ≪1 and under appropriate assumptions on V(x) , our bounds show that T_{c}∼Aexp(B/λ) and Ξ∼Cexp(B/λ) for some explicit coefficients A , B , and C depending on the interaction V(x) and the chemical potential μ . The ratio A/C turns out to be a universal constant, independent of both V(x) and μ . Our analysis is valid for any μ ; for small μ , or low density, our formulas reduce to wellknown expressions involving the scattering length of V(x) .
 Publication:

Physical Review B
 Pub Date:
 May 2008
 DOI:
 10.1103/PhysRevB.77.184517
 arXiv:
 arXiv:0801.4159
 Bibcode:
 2008PhRvB..77r4517H
 Keywords:

 74.20.Fg;
 03.75.Ss;
 67.85.Lm;
 BCS theory and its development;
 Degenerate Fermi gases;
 Condensed Matter  Superconductivity;
 Mathematical Physics
 EPrint:
 RevTeX4, 23 pages. Revised version, to appear in Phys. Rev. B