Invariants of the heat equation

Abstract

Let M be a compact Riemannian manifold without boundary and let P: CV-* CF be a self-adjoint elliptic differential operator with positive definite leading symbol. The asymptotics of the heat equation Tr(exp( -tP)) as t -* 0 + are spectral invariants given by local formulas in the jets of the total symbol of P. Let A(x) and B(x) be polynomials where the degree of B is positive and the leading coefficient is positive. The asymptotics of Tr((P)exp(-B(P))) can be expressed linearly in terms of the asymptotics of Tr(exp(-tP)). Thus no new spectral information is contained in these more general expressions. We also show the asymptotics of the heat equation are genetically non-zero. If one relaxes the condition that the leading symbol of P be definite, the asymptotics of Tr(exp(-P 2 )) and Tr(Pexp(-P 2 )) form a spanning set of invariants. These are related to the zeta and eta functions using the Mellin transform, and a similar non-vanishing result holds except for the single invariant giving the residue of eta at s -0 which vanishes identically.