Effects of confinement for single-well potentials

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.