BCS theory for trapped ultracold fermions

We develop an extension of the well-known BCS-theory to systems with trappe d fermionic atoms. The theory fully includes the quantized energy levels in the trap. The key ingredient is to model the attractive interaction betwee n two atoms by a pseudo-potential which leads to a well defined scattering problem and consequently to a BCS-theory free of divergences. We present nu merical results for the BCS critical temperature and the temperature depend ence of the gap. They are used as a test of existing semi-classical approxi mations.