Study on the general adiabatic expansion theory for heat kernels and its some applications

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We clarified the usefulness and powerfulness of the general adiabatic expansion theory in studying some subjects related to the heat kernels. In particular, based on the theory we obtained a formula for the coefficients of the asymptotic expansion of every derivative of the heat kernel associated to the Riemannian Laplacian. In order to describe the coefficients explicitly up to an arbitrarily high order as universal polynomials built from the curvature, we need only a basic knowledge of calculus added to the formula. (The conventional method requires various kinds of profound knowledge so that only a few coefficients were investigated.) We found also a relation between the adiabatic expansion and certain anomalies which are often referred to by physicists, and introduced an interesting formula for the anomalies.