Local Casimir Energies for an Annulus

The local Casimir energy density for a massless scalar field associated with step-function potentials in a 3+1 dimensional spherical geometry is considered. The potential is chosen to be zero except in an annulus of thickness $\delta$, where it has height $h$, $h\delta=1$. In the limit of zero thickness, an ideal $\delta$-function shell is recovered. The behavior of the energy density as the surface of the annulus is approached is studied in both the strong and weak coupling regimes. The former case corresponds to the well-known Dirichlet shell limit. New results, which shed light on the nature of surface divergences and on the energy contained within the annulus, are obtained in the weak coupling limit. In that case, the energy has a contribution not only from the local energy density, but from an energy term residing entirely on the surface. It is shown that the latter coincides with the integrated local energy density within the annulus. We also study the dependence of local and global quantities on the conformal parameter. In particular new insight is provided on the reason for the divergence in the global Casimir energy in third order