Uniqueness of form extensions and domination of semigroups

TL;DR

This paper introduces a new method using ordered Hilbert spaces and semigroup domination to analyze the uniqueness of form extensions, applicable to magnetic Schrödinger forms on graphs and manifolds.

Contribution

It provides a general abstract result that transfers the uniqueness property from a dominating form to a dominated form, broadening the understanding of form extension uniqueness.

Findings

Established a new method for studying form extension uniqueness

Applied the method to magnetic Schrödinger forms on graphs and manifolds

Demonstrated the transfer of uniqueness properties between forms

Abstract

In this article, we present a new method to study uniqueness of form extensions in a rather general setting. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. Our main abstract result transfers uniqueness of form extension of a dominating form to that of a dominated form. This result can be applied to a multitude of examples including various magnetic Schr\"odinger forms on graphs and on manifolds.