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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.

The electron-phonon interaction lets one electron polarize the ionic lattice and another interact with that delayed deformation. Phonon-mediated electron attraction acts over energy transfers below a characteristic phonon scale. Pairing additionally requires a net attractive interaction in some allowed pairing channel after screened Coulomb repulsion is included. In the conventional weak-coupling situation, retardation separates the low phonon scale from the much larger electronic scale, reducing the effective repulsion. The Cooper instability then makes a Fermi surface unstable to an arbitrarily weak net attraction in the idealized channel; finite temperature cuts off its logarithmic enhancement.

In BCS theory, a momentum-independent pairing amplitude within the cutoff obeys a self-consistency equation containing $\tanh (E/2k_{B}T)/(2E)$, with quasiparticle energy $E=\sqrt{{\xi }^2+|\Delta {|}^2}$. At a continuous transition, take $\Delta \to 0$. A constant effective density of states and coupling combine into $\lambda$, while the band bottom truncates the negative-energy side at $-\sigma$, with $\sigma =\min (\mu ,\theta )$. This gives the supplied linearized gap equation. Its integrand is even, so withit becomes $1/\lambda =[I(\theta ,T_c)+I(\sigma ,T_c)]/2$. If both cutoffs are much larger than $k_B{T}_c$, the given large-cutoff integral gives $I(a,T_c)\simeq \ln (1.13a/(k_B{T}_c))$. Consequently the asymmetric-cutoff BCS transition temperature isFor $\mu >\theta$, this reduces to $1.13\theta e^{-1/\lambda }$; for $0<\mu <\theta$, it is $1.13\sqrt{\theta \mu }{e}^{-1/\lambda }$. The second limit requires $k_B{T}_c\ll \mu$ as well as $k_B{T}_c\ll \theta$. Weak $\lambda$ at fixed nonzero $\mu /\theta$ supplies that limit, but taking $\mu \to 0$ afterward need not preserve it.

For a three-dimensional spin-degenerate parabolic electron band in the degenerate regime,DefineHolding the other model parameters fixed givesThus the log-log slope is $1/3$ below the cutoff-density scale and zero above it. For the specified parameters,This density crossover of a fixed-coupling BCS cutoff model is the requested prescribed model. A real three-dimensional parabolic band also has a changing density of states, so keeping $\lambda$ constant as density varies is an additional imposed assumption, not a consequence of that dispersion.

For noninteracting spinless bosonic pairs, mass $m_B=2m$ and density $n_B=n/2$, homogeneous ideal Bose gas condensation givesUsing the supplied integral $I_B={\int }_0^{\infty }{x}^{1/2}/(e^x-1)dx\simeq 2.3$ gives $C_B=\frac{1}{2}[2/(3I_B){]}^{2/3}\simeq 0.219$. Its slight difference from the more precise $0.218$ is consistent with the rounded integral. Write $a=T_0/(\theta /k_B)=1.13e^{-1/\lambda }\simeq 0.0403$. On the low-density branch,Since $a/C_B<1$, no part of the high-density plateau exceeds $T_B$. The formal crossing condition isThe crossing occurs near $T_c\simeq 7.4\mathrm{K}$ and $T_F\simeq 34\mathrm{K}$.

The apparent low-density excess should not be read as a reliable transition above the ideal condensation scale of preformed pairs. On this extrapolation $k_B{T}_c/\mu =a{u}^{-1/3}$ grows without bound, violating the degenerate-gas and large-lower-cutoff approximations. It reaches one already near $7.5\times 10^{21}{\mathrm{m}}^{-3}$; well before using an arbitrarily small density, the chemical potential, pairing, density of states and pair fluctuations need a self-consistent treatment. The asymptotic formula and ideal-boson curve compare two stipulated models, rather than solving the full BCS-BEC crossover.