Resolvent Trace Asymptotics on Stratified Spaces

TL;DR

This paper develops explicit asymptotic expansions for the resolvent trace of the Hodge-Laplacian on stratified spaces, facilitating the study of spectral invariants and index theory in singular geometric settings.

Contribution

It introduces a functional analytic method to derive resolvent trace asymptotics on stratified spaces, applicable to a broad class of differential operators, without microlocal techniques.

Findings

Established resolvent trace asymptotics for the Hodge-Laplacian on stratified spaces.

Provided a framework for defining spectral invariants like zeta-determinants and analytic torsion.

Method applies inductively to spaces with complex singularities.

Abstract

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of $\Delta$. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.