The resolvent expansion for second order regular singular operators

AbstractThe paper gives a scheme for computing the asymptotics of tr(e−tL) as t → 0+, where L is an elliptic operator of the form L = D2 + x−2A(x) and A(x) is a family of operators satisfying appropriate ellipticity and smoothness conditions. A principle example is the Laplace operator for a manifold with an asymptotically conic singularity. The expansion has the usual terms away from the singularity, appropriately regularized at x = 0, plus singular contributions determined by the ζ-function of (A(0) + 14)12. Applications to index theorems are given in a subsequent paper