= Subtracted grand potential of the reduced BCS model
{title2=$\delta\Omega=V|\Delta|^2/\lambda-\beta^{-1}\sum_{kn}\ln[(\omega_n^2+E_k^2)/(\omega_n^2+\xi_k^2)]$}
Subtracting the normal-state <grand potential> fixes the reference constant in the uniform <BCS pairing-field effective action>. Here $\omega_n=(2n+1)\pi/\beta$ has energy units and $E_k^2=\xi_k^2+|\Delta|^2$. The ratio cancels a field-independent determinant phase and produces a convergent high-frequency sum under the same momentum regulator. Differentiation gives the <BCS gap equation>. For a fixed attractive pairing shell, this is a finite regulated model; an unrestricted continuum contact interaction additionally needs ultraviolet renormalization. Its weak-coupling condensation density $-\nu_F\Delta^2/2$ uses a single-spin <density of states> per volume.
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