A discrete leading symbol and spectral asymptotics for natural differential operators

TL;DR

This paper introduces a new approach to analyze natural differential operators in Riemannian geometry with non-Laplace leading symbols, focusing on their spectral asymptotics and applications to fundamental solutions and conformally covariant operators.

Contribution

It defines a discrete leading symbol for these operators and demonstrates how to compute it pointwise and from spectral asymptotics, advancing understanding of their spectral properties.

Findings

Defined a discrete leading symbol for non-Laplace type operators

Established methods to compute spectral asymptotics from this symbol

Applied the framework to fundamental solutions and conformally covariant operators

Abstract

We initiate a systematic study of natural differential operators in Riemannian geometry whose leading symbols are not of Laplace type. In particular, we define a discrete leading symbol for such operators which may be computed pointwise, or from spectral asymptotics. We indicate how this can be applied to the computation of another kind of spectral asymptotics, namely asymptotic expansions of fundamental solutions, and to the computation of conformally covariant operators.