We study a model of a multiband Fermi-surface structure to investigate its effect on the superconducting transition temperature in the limit of a high number of bands. We consider a simple limit consisting of an infinite number of identical locally pairwise coupled bands with intraband and interband hopping and a multiband generalized Bardeen-Cooper-Schrieffer Hamiltonian. The self-consistent mean-field system of equations which determines the intraband and interband order parameters decouples to two independent equations, unless the interband hopping integral is non-zero, in which case an energetically stable superconducting phase appears, where both the intraband and the interband gaps are non-zero. We demonstrate that for all values of the interband coupling constant the critical transition temperature is enhanced compared with the pure intraband critical transition temperature. The model is equivalent to a multiple momentum exchange originating from the interband coupling and thus modelling a highly anisotropic gap structure.
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