Formulas for Hadamard coefficients in terms of Green's operators

TL;DR

This paper explores methods to derive Hadamard coefficients from Green's operators for hyperbolic equations, enhancing understanding of their relationship through resolvents, powers, and manifold products.

Contribution

It introduces new techniques to extract Hadamard coefficients from Green's operators, including using resolvents, powers, and manifold products.

Findings

Methods to obtain Hadamard coefficients from Green's operators

Use of resolvent and powers of Green's operators

Analysis of manifold products to facilitate the derivation

Abstract

We describe various ways of obtaining the Hadamard coefficients associated to a normally hyperbolic operator from the corresponding Green's operators. As the Hadamard expansion on its own is not enough for this, we include additional information either by considering something like a resolvent or powers of Green's operators or by looking at a product of the original manifold with the real line.