Criticality of Schr\"{o}dinger forms and recurrence of Dirichlet forms

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