A dynamic Jahn-Teller theory for high-T_c superconductivity

The Jahn-Teller effect gives rise to a negative McMillan coupling constant λ. This negative λ gives rise to oscillatory solutions of the BCS gap function Δ(E). The period of oscillations is given (approximately) by 4 3ω, where ω is the phonon or exciton frequency. These oscillations require an Einstein spectrum, or at least small dispersion. The oscillatory solutions are insensitive to the (non-resonant) Coulomb repulsion μ. They are obtained from the Dyson equation, which serves as a starting point of the Eliashberg theory, but not from the Eliashberg equation per se. We suggest that the singular nature of these solutions requires a minor modification of Eliashberg's derivation. These solutions are particularly favorable for excitonic superconductivity, for which the range of interaction η= 3 2πv_F/ω is one, or a few, lattice constants, and the Cooper pairs have some "boson" character. We suggest that these solutions arise because the conditions required for the validity of Migdal's theorem do not hold.