Pseudodifferential operators on manifolds: a coordinate-free approach

TL;DR

This paper reviews coordinate-free calculi of pseudodifferential operators on manifolds and applies them to analyze spectral projection behavior under coefficient perturbations.

Contribution

It introduces a coordinate-free calculus for pseudodifferential operators and demonstrates its use in deriving new spectral projection results for elliptic operators.

Findings

New results on spectral projections under perturbations

Development of a coordinate-free calculus for pseudodifferential operators

Enhanced understanding of operator behavior on manifolds

Abstract

This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a self-adjoint elliptic second order differential operator under perturbation of coefficients.