The BCS - BEC Crossover In Arbitrary Dimensions

Cold atom traps and certain neutron star layers may contain fermions with separation much larger than the range of pair-wise potentials yet much shorter than the scattering length. Such systems can display {\em universal} characteristics independent of the details of the short range interactions. In particular, the energy per particle is a fraction $\xi$ of the Fermi energy of the free Fermion system. Our main result is that for space dimensions D smaller than two and larger than four a specific extension of this problem readily yields $\xi=1$ for all $D \le 2$ whereas $\xi$ is rigorously non-positive (and potentially vanishing) for all $ D \ge 4$. We discuss the D=3 case. A particular unjustified recipe suggests $\xi=1/2$ in D=3.