  We show that the quantum dynamics of a thin spherical shell in general
relativity is equivalent to the Coulomb-Dirac equation on the half line. The
Hamiltonian has a one-parameter family of self-adjoint extensions with a
discrete energy spectrum $|E| < m$, and a continuum of scattering states for
$|E|>m$, where $m$ is the rest mass of the shell and $E$ is the
Arnowitt-Deser-Misner mass. For sufficiently large $m$, the ground state energy
level is negative. This suggests that classical positivity of energy does not
survive quantization. The scattering states provide a realization of
singularity avoidance. We speculate on the consequences of these results for
black hole radiation.
