A DeWitt expansion of the heat kernel for manifolds with a boundary

Abstract

The DeWitt (DW) expansion of the heat kernel G_Delta (x, x'; tau ), for a second order elliptic differential operator Delta is extended to manifolds with a boundary. The boundary conditions are satisfied by including an additional term to the usual DW contribution appropriate for manifolds without a boundary, which provides an asymptotic expansion as tau to 0 for x approximately=x'. The extra piece is based on geodesic paths linking x and x', which undergo reflection on the boundary, just as the DW expression is based on the direct geodesic from x to x'. The boundary contributions to Tr(e^{- tau Delta} ), which are an expansion in tau ¹2/, are found in agreement with previous indirect methods, although the corresponding contributions to G_Delta involve non-polynomial functions of tau . Conservation equations for vector and tensor fields obtained from the heat kernel, which include terms restricted to the boundary, are also verified. The results are applied to determine the leading singular behaviour of the Green function for Delta at x=x' in the neighbourhood of the boundary.