= Asymmetric-cutoff BCS transition temperature
{title2=$k_BT_c\simeq1.13\sqrt{\theta\sigma}\,e^{-1/\lambda}$}
For a constant effective <density of states> and attractive coupling $\lambda$, let the linearized <BCS theory> gap equation integrate $\tanh(\xi/(2k_BT_c))/(2\xi)$ over $-\sigma<\xi<\theta$, with positive cutoffs. Its even integrand makes the result the mean of two positive-cutoff logarithms. When $k_BT_c\ll\sigma,\theta$, $1/\lambda\simeq\frac12\ln(1.13\theta/(k_BT_c))+\frac12\ln(1.13\sigma/(k_BT_c))$, giving the displayed geometric mean of cutoffs.
If a band bottom truncates the lower side, $\sigma=\min(\mu,\theta)$ for positive <chemical potential> $\mu$. The weak-coupling limit at fixed $\mu/\theta>0$ must not be interchanged with an uncontrolled $\mu\to0$ extrapolation: the lower logarithm requires $\mu\gg k_BT_c$ when $\mu<\theta$.
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