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Criticality for Schrödinger type operators based on recurrent symmetric stable processes


Author: Masayoshi Takeda
Journal: Trans. Amer. Math. Soc. 368 (2016), 149-167
MSC (2010): Primary 60J45; Secondary 60J75, 31C25
DOI: https://doi.org/10.1090/tran/6319
Published electronically: April 3, 2015
MathSciNet review: 3413859
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Abstract: Let $ \mu $ be a signed Radon measure on $ \mathbb{R}^1$ in the Kato class and consider a Schrödinger type operator $ \mathcal {H}^{\mu }=(-d^2/dx^2)^{\frac {\alpha }{2}} + \mu $ on $ \mathbb{R}^1$. Let $ 1\leq \alpha <2$ and suppose the support of $ \mu $ is compact. We then construct a bounded $ \mathcal {H}^{\mu }$-harmonic function uniformly lower-bounded by a positive constant if $ \mathcal {H}^{\mu }$ is critical. Moreover, we show that there exists no bounded positive $ \mathcal {H}^{\mu }$-harmonic function if $ \mathcal {H}^{\mu }$ is subcritical.


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Additional Information

Masayoshi Takeda
Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan
Email: takeda@math.tohoku.ac.jp

DOI: https://doi.org/10.1090/tran/6319
Received by editor(s): July 24, 2013
Received by editor(s) in revised form: October 24, 2013
Published electronically: April 3, 2015
Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research (No. 22340024 (B), Japan Society for the Promotion of Science.
Article copyright: © Copyright 2015 American Mathematical Society