Full Expansion of the Heat Trace for Cone Differential Operators

The operator e−tA and the heat trace Tr e−tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0+. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A− λ)−1. The main step will consist in approximating this operator family by a parametrix to A− λ using a suitable parameter–dependent calculus. Introduction In this paper the operator e for t > 0 is investigated on manifolds with conical singularities. The operator A is assumed to be an elliptic differential operator of arbitrary positive order, not necessarily self–adjoint, but satisfying an analog of Agmon’s condition (parameter–ellipticity) in a sector {λ ∈ C | 0 < φ0 < | arg(λ − c0)| ≤ π} for some π/2 > φ0 > 0 and c0 > 0. Our aim is to describe in a precise way the resolvent (A − λ) for |λ| → ∞ as well as the operator e (heat operator) and its trace Tr e (heat trace) as t → 0. From the analytic point of view a cone is a product (0, c) × X together with a metric of the form dr + rgX(r), where gX(r) is a smooth family of Riemannian metrics on the ‘cone base’ X. Here, X is assumed to be a smooth compact manifold without boundary. For this reason, the analysis on a manifold with conical singularities takes place on a manifold with boundary B with the mentioned product structure near ∂B = X. The natural differential ∗This work was supported by Max-Planck-Gesellschaft, Bonn