Let M be a compact
Riemannian manifold without boundary and let P : C∞V → C∞V 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(A(P)exp(−tB(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(−tP2)) and Tr(Pexp(−tP2)) 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.
