Magnetic fields on resistance spaces

TL;DR

This paper studies magnetic energy forms on resistance spaces, establishing conditions for their mathematical well-posedness without requiring energy dominance, thus advancing the understanding of magnetic operators in complex geometric settings.

Contribution

It introduces new criteria for the closability and self-adjointness of magnetic energy forms on resistance spaces, broadening the theoretical framework for magnetic operators.

Findings

Provided sufficient conditions for closability.

Established criteria for self-adjointness.

Applied geometric conditions on the measure.

Abstract

On a metric measure space $X$ that supports a regular, strongly local resistance form we consider a magnetic energy form that corresponds to the magnetic Laplacian for a particle confined to $X$. We provide sufficient conditions for closability and self-adjointness in terms of geometric conditions on the reference measure without assuming energy dominance.