Criticality of Schrödinger forms and recurrence of Dirichlet forms

Introducing the notion of extended Schrödinger spaces, we define criticality and subcriticality forms in manner similar to recurrence transience Dirichlet forms. We show that a form is critical (resp. subcritical) if only there exists an excessive function associated semigroup defined by h h -transform recurrent transient). give analytical condition for terms bottom spectrum. introduce subclass alttext="script upper K Subscript H"> K H encoding="application/x-tex">{\mathcal {K}}_H local Kato class with potential critical. Critical lead us Hardy-type inequalities. As example, treat fractional operators reconsider classical Hardy inequality our approach.