Criticality of Schr�dinger forms and recurrence of Dirichlet forms
M Takeda, T Uemura�- Transactions of the American Mathematical Society, 2023 - ams.org
M Takeda, T Uemura
Transactions of the American Mathematical Society, 2023•ams.orgIntroducing the notion of extended Schr�dinger spaces, we define the criticality and
subcriticality of Schr�dinger forms in the manner similar to the recurrence and transience of
Dirichlet forms. We show that a Schr�dinger form is critical (resp. subcritical) if and only if
there exists an excessive function of the associated Schr�dinger 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 Schr�dinger forms in terms of the bottom�…
subcriticality of Schr�dinger forms in the manner similar to the recurrence and transience of
Dirichlet forms. We show that a Schr�dinger form is critical (resp. subcritical) if and only if
there exists an excessive function of the associated Schr�dinger 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 Schr�dinger forms in terms of the bottom�…
Abstract
Introducing the notion of extended Schr�dinger spaces, we define the criticality and subcriticality of Schr�dinger forms in the manner similar to the recurrence and transience of Dirichlet forms. We show that a Schr�dinger form is critical (resp. subcritical) if and only if there exists an excessive function of the associated Schr�dinger semigroup and the Dirichlet form defined by -transform of the excessive function is recurrent (resp. transient). We give an analytical condition for the subcriticality of Schr�dinger forms in terms of the bottom of spectrum.
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