  Doped strontium titanate becomes superconducting at a density as low as n = 5
x 10^17 cm^-3, where the Fermi energy is orders of magnitude smaller than the
longitudinal-optical-phonon frequencies. In this limit the only optical mode
with a frequency which is smaller than the Fermi energy is the plasmon. In
contrast to metals, the interaction strength is weak due to screening by the
crystal, which allows the construction of a controllable theory of plasmon
superconductivity. We show that plasma mediated pairing alone can account for
the observed transition temperatures if the screening by the crystal is reduced
in the slightly doped samples compared with the insulating ones. This mechanism
can also explain the pairing in the two-dimensional superconducting states
observed at surfaces and interfaces with other oxides. We also discuss unique
features of the plasmon mechanism, which appear in the tunneling density of
states above the gap.
