Criticality and subcriticality of generalized Schrödinger forms
Let \((\mathcal{E},D(\mathcal{E}))\) be a regular symmetric Dirichlet form on \(L^2(X;m)\) and \(\mu=\mu^+-\mu^-\) be a suitable signed Radon measure such that the positive (resp. negative) part \(\mu^+\) (resp. \(\mu^-\)) belongs to the local Kato class (resp. the Kato class). The Schrödinger form \((\mathcal{E}^{\mu},D(\mathcal{E}^{\mu}))\) is defined by \[ \mathcal{E}^{\mu}(u,u)=\mathcal{E}(u,u)+\int_Xu^2d\mu,\quad u\in D(\mathcal{E}^{\mu})=D(\mathcal{E})\cap L^2(X;\mu^+). \] The author defines criticality or subcriticality for \(\mathcal{E}^{\mu}\) through the \(h\)-transform. In Section 2, several properties of Schrödinger forms are presented, where the extended Schrödinger space \(\mathcal{D}_e(\mathcal{E}^{\mu})\) is defined by the inverse \(h\)-transform. In Section 3, several Poincaré-type inequalities for Schrödinger forms are given. In Section 4, criticality, subcriticality and supercriticality for \(\mathcal{E}^{\mu}\) are defined. In the final section, for a certain potential \(\mu\), an analytic characterization of the criticality, subcriticality and supercriticality for \(\mathcal{E}^{\mu}\) is given in terms of the bottom of the spectrum. In a word, the paper extends the results in [\textit{Y. Pinchover} and \textit{K. Tintarev}, J. Funct. Anal. 230, No. 1, 65--77 (2006; Zbl 1086.35025)] to more general Dirichlet forms with non-local part.
scientific article