  We study bound states generated by a unique potential minimum in the
situation where the system is strongly confined to a bounded region containing
the minimum (by imposing Dirichlet boundary conditions). In this case the
eigenvalues of the confined system differ from those of the unconfined system
by an exponentially small quantity in the semiclassical limit. An asymptotic
expansion for this shift is established. The formulas are evaluated explicitly
for the harmonic oscillator and an application to the Coulomb potential at a
fixed angular momentum is given.
