The BCS 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 $\cosh ({\delta }_{\mu }\mathrm{/}T)\le 2$, 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 \cite{FHNS,HHSS,HS}. For $\cosh ({\delta }_{\mu }\mathrm{/}T)>2$ 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.