  In a recent study of the self-adjoint extensions of the Hamiltonian of a
particle confined to a finite region of space, in which we generalized the
Heisenberg uncertainty relation to a finite volume, we encountered bound states
localized at the wall of the cavity. In this paper, we study this situation in
detail both for a free particle and for a hydrogen atom centered in a spherical
cavity. For appropriate values of the self-adjoint extension parameter, the
bound states lo calized at the wall resonate with the standard hydrogen bound
states. We also examine the accidental symmetry generated by the Runge-Lenz
vector, which is explicitly broken in a spherical cavity with general Robin
boundary conditions. However, for specific radii of the confining sphere, a
remnant of the accidental symmetry persists. The same is true for an electron
moving on the surface of a finite circular cone, bound to its tip by a 1/r
potential.
