HEAT-KERNEL APPROACH TO THE ZETA-FUNCTION REGULARIZATION OF THE CASIMIR ENERGY FOR DOMAINS WITH CURVED BOUNDARIES

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Abstract
We combine a zeta-function definition and heat-kernel series to derive Casimir energy expansions parametrizing the UV divergences in the presence of arbitrarily shaped smooth boundaries. Their terms, in the form of a geometrical object times a divergence, allow for drawing conclusions on the scale dependence and on the finiteness of the vacuum energy when limiting surfaces have been introduced. Different behaviors are found depending, among other factors, on the even or odd character of the space dimension. A number of controversial points are cleared up and some misstatements in the literature are properly rigorized.

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