The gap equation for spin-polarized fermions

We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For cosh (delta(mu)/T) <= 2, with T the temperature and delta(mu) the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in [Frank, R. L., Hainzl, C., Naboko, S., and Seiringer, R., J., Geom. Anal. 17, 559-567 (2007); Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J. P., Commun., Math. Phys. 281, 349-367 (2008); and Hainzl, C. and Seiringer, R., Phys. Rev. B 77, 184517-110 435 (2008)]. For cosh (delta(mu)/T) > 2 the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit. C (C) 2012 American Institute of Physics. [doi: 10.1063/1.3670747]