  Superconductivity of doped SrTiO3 is proven to be a particular case of the
broader concept of the non-adiabatic pairing mediated by phonons with frequency
comparable or larger the Fermi energy. We argue that, for carrier
concentrations exceeding that of the mobility edge, the superconductivity of
doped SrTiO3 is mediated by interaction of electrons with several longitudinal
(LO) optical polar phonons. The electronic spectrum of SrTiO3 consists at low
temperatures of three conduction bands which are successively doped. Each band
contributes to the Cooper instability and exhibits a superconducting gap in the
energy spectrum. The theory presented below predicts maxima in dependence of
Tc(n)-the transition temperature on n, the number of electrons owing to the
following mechanism. Doping by electrons increases density of states at the
Fermi surface and Tc initially grows up. At the same time, screening on the
part of accumulating charges tends to reduce amplitude of the electrical fields
inherent in LO phonon modes and at larger concentrations the matrix element of
interaction between electrons and LO phonons decreases. The compromise between
the two tendencies leads to maxima in the Tc(n)-dependence providing
interpretation to one of the most intriguing experimental findings in Xiao Lin
et al [Phys. Rev. Lett. 112, 207002 (2014)]. Having reached a maximum in the
third band, the superconducting transition finally decreases, rounding out the
Tc(n)-dome, the three maxima in Tc(n)with accompanying superconducting gaps
emerging consecutively as electrons fill successive bands. This arises from
attributes of the LO optical phonon pairing mechanism. More generally, the
mechanism opens prospect of increasing temperature of the superconducting
transition in transition-metals oxides and other polar crystals.
