Cooper pair dispersion relation for weak to strong coupling

Cooper pairing in two dimensions is analyzed with a set of renormalized equ ations to determine its binding energy for any fermion number density and a ll coupling assuming a,generic pairwise residual interfermion interaction. Also considered are Cooper pairs (CP's) with nonzero center-of-mass momentu m (CMM) and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dom inates the pair excitation energy in weak coupling (also called the BCS reg ime) while the more familiar quadratic term prevails in strong coupling (th e Bose regime). The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a mo del of superconductivity as a Bose-Einstein condensation of CPs where the t ransition temperature vanishes for all dimensionality d less than or equal to 2 for quadratic dispersion, but is nonzero for all d greater than or equ al to 1 for linear dispersion.