  The idealized theory of quantum vacuum energy density is a beautiful
application of the spectral theory of differential operators with boundary
conditions, but its conclusions are physically unacceptable. A more plausible
model of a reflecting boundary that stays within linear spectral theory
confines the waves by a steeply rising potential function, which can be taken
as a power of one coordinate, z^\alpha. We report investigations of this model
with considerable student involvement. An exact analytical solution with some
numerics for \alpha=1 and an asymptotic (semiclassical) analysis of a related
problem for \alpha=2 are presented.
