  In a previous work we obtained exact solutions for the proper time quantum
mechanics of a thin dust shell, collapsing in a vacuum. We extend these results
to the quantum collapse of a dust shell surrounding a pre-existing black hole.
In lieu of exact solutions, which have so far proved difficult to obtain for
this system, we establish the essential features of the quantum shell through a
Wentzel-Kramers-Brillouin (WKB) approximation, which is valid only when the
mass of the shell is much greater than the Planck mass. There are many
similarities with the vacuum collapse: only bound states exist and the proper
energy spectrum of the shell is unaffected by the presence of the central black
hole to this order. There are no peculiar or distinguishing features of the
wave function near the black hole horizon. It vanishes at the center and
oscillates between the origin and the classically forbidden region, beyond
which it decays exponentially.
