Abstract
We investigate the appearance of superconductivity in a model with Peierls electron-phonon coupling for very low carrier concentrations. Superconductivity with very high critical temperatures is found to occur at all carrier concentrations if the electron-phonon coupling is not too weak, with a gap of $s+{s}^{*}$ symmetry that closes on the Fermi surface for appropriate parameters. The possibility of finding the elusive $p$-type superconductivity is also revealed, although this turns out not to be stable within the approximations we use.
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