Uniqueness of form extensions and domination of semigroups

TL;DR

This paper investigates the conditions under which form extensions are unique for magnetic Schrödinger forms, utilizing the theory of ordered Hilbert spaces and semigroup domination, with applications to graphs and Euclidean domains.

Contribution

It introduces a new abstract characterization of form domination and transfers uniqueness properties from dominating to dominated forms in magnetic Schrödinger contexts.

Findings

Established a characterization of form domination in abstract Hilbert space setting.

Proved a transfer theorem for uniqueness of form extensions.

Applied results to magnetic Schrödinger forms on graphs and Euclidean domains.

Abstract

In this article, we study questions of uniqueness of form extension for certain magnetic Schr\"odinger forms. The method is based on the theory of ordered Hilbert spaces and the concept of domination of semigroups. We review this concept in an abstract setting and give a characterization in terms of the associated forms. Then we use it to prove a theorem that transfers uniqueness of form extension of a dominating form to that of a dominated form. This result is applied in two concrete situations: magnetic Schr\"odinger forms on graphs and on domains in Euclidean space.