Green functions of higher-order differential operators

TL;DR

This paper investigates the Green functions of higher-order differential operators on Riemannian manifolds using heat kernel methods, providing asymptotic expansions and criteria related to the Huygens principle.

Contribution

It introduces a detailed analysis of Green functions for higher-order operators formed from second-order Laplace type operators, linking their singularities to heat kernel coefficients.

Findings

Asymptotic expansion of Green functions near the diagonal in any dimension.

A simple criterion for the validity of the Huygens principle.

Expression of singularities of Green functions in terms of heat kernel coefficients.

Abstract

The Green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian manifold are investigated via the heat kernel methods. We study the resolvent of a special class of higher-order operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric but with different bundle connections and potential terms. The asymptotic expansion of the Green functions near the diagonal is studied in detail in any dimension. As a by-product a simple criterion for the validity of the Huygens principle is obtained. It is shown that all the singularities as well as the non-analytic regular parts of the Green functions of such high-order operators are expressed in terms of the usual heat kernel coefficients $a_k$ for a special Laplace type second-order operator.