  We use scanning tunneling spectroscopy to investigate the filled and empty
electronic states of superconducting single-unit-cell FeSe deposited on
SrTiO$_3$(001). We map the momentum-space band structure by combining
quasiparticle interference imaging with decay length spectroscopy. In addition
to quantifying the filled-state bands, we discover a $\Gamma$-centered electron
pocket 75 meV above the Fermi energy. Our density functional theory
calculations show the orbital nature of empty states at $\Gamma$ and suggest
that the Se height is a key tuning parameter of their energies, with broad
implications for electronic properties.
