Casimir Energy of a Scalar Field with a Spacedependent Mass Distribution*

AEBTFLWT The Casimir energy is evaluated for a free scalar field that has a mass term m2(q), depending on one space coordinate 21. The formalism for evaluating the Casimir energy is developed for the case of m2(q) finite everywhere in d-dimensional space-time. The case with m2(q) = m$9(L/2-1x11) + m&O(]q]-L/2) is explicitly evaluated for any value of mo and moo without any approximation. The result consists of volume energy terms, a surface energy term, and a non-leading term. Most of the W divergences are in the volume energy terms and renormalize the coupling constants of the underlying theory. The surface energy term is finite for d 5 4 and divergent for d > 5 due to the boundaries being sharp. A closed finite expression is obtained for the non-leading term. Our results are shown to reproduce the known Casimir energies for the limiting cases, mo + 00 and moo-+ co. 2