Functional derivatives of Tc for a two-band superconductor: application to MgB2

We address the question of how phonons of various frequencies contribute to the superconducting transition temperature $T_c$ of a multi-band superconductor by considering the functional derivatives of $T_c$ with respect to various intraband and interband electron-phonon coupling functions. A general scheme for computing such functional derivatives is developed. The proofs are given that the functional derivatives which are diagonal in band indices are linear in phonon energy $\mathrm{\Omega }$ at small $\mathrm{\Omega }$, while the functional derivatives which are off-diagonal in band indices diverge at $\mathrm{\Omega }$ = 0 as 1/$\mathrm{\Omega }$. The case of a two-band model for MgB${}_2$ is treated numerically.