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

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.