  Superconductivity in one-atom-layer iron selenide (FeSe) on a strontium
titanate (STO) substrate is enhanced by almost an order of magnitude with
respect to bulk FeSe. There is recent experimental evidence suggesting that
this enhancement persists in FeSe/STO nano-islands. More specifically, for
sizes $L \sim 10$ nm, the superconducting gap is a highly non-monotonic
function of $L$ with peaks well above the bulk gap value. This is the expected
behavior only for weakly-coupled metallic superconductors such as Al or Sn.
Here we develop a theoretical formalism to describe these experiments based on
three ingredients: Eliashberg theory of superconductivity in the weak coupling
limit, pairing dominated by forward scattering and periodic orbit theory to
model spectral fluctuations. We obtain an explicit analytical expression for
the size dependence of the gap that describes quantitatively the experimental
results with no free parameters. This is a strong suggestion that
superconductivity in FeSe/STO is mediated by STO phonons. We propose that,
since FeSe/STO is still a weakly coupled superconductor, quantum size effects
can be used to further enhance the bulk critical temperature in this interface.
