  SrTiO$_3$ exhibits superconductivity for carrier densities $10^{19}-10^{21}$
cm$^{-3}$. Across this range, the Fermi level traverses a number of vibrational
modes in the system, making it ideal for studying dilute superconductivity. We
use high-resolution planar-tunneling spectroscopy to probe chemically-doped
SrTiO$_3$ across the superconducting dome. The over-doped superconducting
boundary aligns, with surprising precision, to the Fermi energy crossing the
Debye energy. Superconductivity emerges with decreasing density, maintaining
throughout the Bardeen-Cooper-Schrieffer (BCS) gap to transition-temperature
ratio, despite being in the anti-adiabatic regime. At lowest superconducting
densities, the lone remaining adiabatic phonon van Hove singularity is the soft
transverse-optic mode, associated with the ferroelectric instability. We
suggest a scenario for pairing mediated by this mode in the presence of
spin-orbit coupling, which naturally accounts for the superconducting dome and
BCS ratio.
