Singularities of Green functions of the products of the Laplace type operators

TL;DR

This paper investigates the singularity structure of Green functions for products of Laplace-type operators on Riemannian manifolds, providing explicit formulas in terms of heat kernel coefficients.

Contribution

It introduces explicit formulas for Green function singularities of product operators of Laplace type, linking them to heat kernel coefficients.

Findings

Derived explicit formulas for Green function singularities

Connected singularities to heat kernel coefficients

Analyzed operators formed by products of Laplace-type operators

Abstract

The structure of diagonal singularities of Green functions of partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian man ifold is studied. A special class of operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric and a unique bundle connection but with different potential terms is investigated. Explicit simple formulas for singularities of Green functions of such operators in terms of the usual heat kernel coefficients are obtained.