Trace expansions for elliptic cone operators with stationary domains

TL;DR

This paper provides a detailed asymptotic analysis of the trace of the resolvent for elliptic cone operators with stationary domains, distinguishing between minimal extension and finite-rank components.

Contribution

It offers a full asymptotic expansion of the resolvent's trace components for stationary domains, advancing understanding of elliptic cone operators' spectral behavior.

Findings

Asymptotic expansion of the resolvent trace for stationary domains

Decomposition of the resolvent into minimal extension and finite-rank parts

Development of techniques for analyzing elliptic cone operators

Abstract

We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand the component of finite rank in the case where the domain is stationary. The results make use, and develop further, our previous investigations on the analytic and geometric structure of the resolvent. The analysis of nonstationary domains, considerably more intricate, is pursued elsewhere.