Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

TL;DR

This paper investigates the essential self-adjointness of first-order differential operators with low-regularity metrics on non-compact manifolds, establishing conditions under which their powers are also essentially self-adjoint, with applications to Dirac operators.

Contribution

It provides new criteria linking low-regularity metrics and boundary properties to essential self-adjointness of operators and their powers, extending previous results to less regular settings.

Findings

Essential self-adjointness is equivalent to a negligible boundary property.

Higher powers are essentially self-adjoint under regularity conditions.

Applications include Dirac operators on non-smooth metrics.

Abstract

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth.