Dynamics on Grassmannians and resolvents of cone operators

TL;DR

This paper investigates the asymptotic behavior of the resolvent trace of elliptic cone operators on manifolds with boundary, revealing structural insights through Grassmannian dynamics under minimal symbol conditions.

Contribution

It establishes the existence and detailed structure of the resolvent trace asymptotics for general elliptic cone operators with minimal assumptions on symbols.

Findings

Asymptotic expansion of the resolvent trace is proven to exist.

The structure of the asymptotics is elucidated through Grassmannian flow analysis.

Minimal symbol conditions suffice for the main results.

Abstract

The paper proves the existence and elucidates the structure of the asymptotic expansion of the trace of the resolvent of a closed extension of a general elliptic cone operator on a compact manifold with boundary as the spectral parameter tends to infinity. The hypotheses involve only minimal conditions on the symbols of the operator. The results combine previous investigations by the authors on the subject with an analysis of the asymptotics of a family of projections related to the domain. This entails a fairly detailed study of the dynamics of a flow on the Grassmannian of domains.