Criticality of Schr\"{o}dinger Forms and Recurrence of Dirichlet Forms

TL;DR

This paper introduces a new framework for understanding the criticality and subcriticality of Schr"odinger forms, linking these concepts to recurrence and transience of Dirichlet forms, and applies it to fractional Schr"odinger operators with Hardy potentials.

Contribution

It defines criticality and subcriticality for Schr"odinger forms using extended Schr"odinger spaces and establishes their equivalence to recurrence and transience via $h$-transform, including applications to Hardy potentials.

Findings

Established a sufficient condition for subcriticality based on spectrum bottom.

Proved Schr"odinger forms with certain Hardy potentials are always critical.

Connected criticality/subcriticality to recurrence/transience through $h$-transform.

Abstract

Introducing the notion of extended Schr\"odinger spaces, we define the criticality and subcriticality of Schr\"odinger forms in the same manner as the recurrence and transience of Dirichlet forms, and give a sufficient condition for the subcriticality of Schr\"odinger forms in terms the bottom of spectrum. We define a subclass of Hardy potentials and prove that Schr\"odinger forms with potentials in this subclass are always critical, which leads us to optimal Hardy type inequality. We show that this definition of criticality and subcriticality is equivalent to that there exists an excessive function with respect to Schr\"odinger semigroup and its generating Dirichlet form through $h$-transform is recurrent and transient respectively. As an application, we can show the recurrence and transience of a family of Dirichlet forms by showing the criticality and subcriticaly of Schr\"odinger forms and show the other way around through $h$-transform, We give a such example with fractional Schr\"odinger operators with Hardy potential.