Criticality and subcriticality of generalized Schrödinger forms
type: Item
instance of: scholarly article
title: Criticality and subcriticality of generalized Schrödinger forms
zbMATH Open document ID: 1317.31019
author: Masayoshi Takeda
published in: Illinois Journal of Mathematics
publication date: 2015-04-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1427897177
review text: 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.
reviewed by: Ze-Chun Hu
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 6428030
zbMATH Keywords: Dirichlet form
zbMATH Keywords: generalized Schrödinger form
zbMATH Keywords: \(h\)-transform
MaRDI profile type: Publication
cites work: Gaugeability and conditional gaugeability
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cites work: A ground state alternative for singular Schrödinger operators
cites work: Perturbation of Dirichlet forms by measures
cites work: A variational formula for Dirichlet forms and existence of ground states
cites work: Differentiability of spectral functions for symmetric $\alpha$-stable processes
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description: scientific article