On the domain of Dirac and Laplace type operators on stratified spaces

TL;DR

This paper establishes the essential self-adjointness and domain characterization of Dirac operators on stratified spaces with cone-edge metrics, using an explicit, microlocal-free approach based on interpolation scales.

Contribution

It provides a new, explicit method to identify domains of Dirac operators on stratified spaces without microlocal analysis, under spectral Witt conditions.

Findings

Proves essential self-adjointness of Dirac operators on stratified spaces.

Identifies domains with weighted edge Sobolev spaces.

Applicable to Gauss-Bonnet and spin Dirac operators.

Abstract

We consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition.