Criticality and subcriticality of generalized Schr\"odinger forms with non-local perturbations

TL;DR

This paper extends the concepts of criticality, subcriticality, and supercriticality to non-local Schr"odinger forms within the framework of quasi-regular Dirichlet forms, providing an analytic spectral characterization.

Contribution

It generalizes Takeda's definitions to non-local cases and offers a spectral analysis approach for these classifications.

Findings

Extended definitions of criticality to non-local forms

Provided spectral characterization of criticality levels

Enhanced understanding of Schr"odinger forms with non-local perturbations

Abstract

In this paper, we shall treat the Schr\"odinger forms with non-local perturbations. We first extend the definitions of subcriticality, criticality and supercriticality for the Schr\"odinger forms by Takeda in [IJM, 2015] to the non-local cases in the context of quasi-regular Dirichlet forms. Then we prove an analytic characterization of these definitions via the bottom of the spectrum set.