Density crossover of a fixed-coupling BCS cutoff model ($n_{\theta }=(2m\theta /{\hslash }^2{)}^{3/2}/(3{\pi }^2)$)

Set the chemical potential equal to the degenerate three-dimensional spin-degenerate Fermi energy, proportional to $n^{2/3}$, and keep the effective coupling and upper cutoff fixed. The asymmetric-cutoff BCS transition temperature scales as $n^{1/3}$ below $n_{\theta }$ and saturates above it. At low density the predicted $k_B{T}_c/\mu$ instead grows as $n^{-1/3}$, eventually invalidating the degenerate and logarithmic approximations. This is a prescribed cutoff model, not a self-consistent treatment of the BCS-BEC crossover.