  We investigate the simple harmonic oscillator in a 1-d box, and the 2-d
isotropic harmonic oscillator problem in a circular cavity with perfectly
reflecting boundary conditions. The energy spectrum has been calculated as a
function of the self-adjoint extension parameter. For sufficiently negative
values of the self-adjoint extension parameter, there are bound states
localized at the wall of the box or the cavity that resonate with the standard
bound states of the simple harmonic oscillator or the isotropic oscillator. A
free particle in a circular cavity has been studied for the sake of comparison.
This work represents an application of the recent generalization of the
Heisenberg uncertainty relation related to the theory of self-adjoint
extensions in a finite volume.
