  We revisit the old black hole S-Matrix construction and its new partial wave
expansion of 't Hooft. Inspired by old ideas from non-critical string theory &
$c=1$ Matrix Quantum Mechanics, we reformulate the scattering in terms of a
quantum mechanical model\textemdash of waves scattering off inverted harmonic
oscillator potentials\textemdash that exactly reproduces the unitary black hole
S-Matrix for all spherical harmonics; each partial wave corresponds to an
inverted harmonic oscillator with ground state energy that is shifted relative
to the s-wave oscillator. Identifying a connection to 2d string theory allows
us to show that there is an exponential degeneracy in how a given total initial
energy may be distributed among many partial waves of the 4d black hole.
