Criticality for Schrödinger type operators based on recurrent symmetric stable processes
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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.References
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Additional Information
- Masayoshi Takeda
- Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan
- MR Author ID: 211690
- Email: takeda@math.tohoku.ac.jp
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 149-167
- MSC (2010): Primary 60J45; Secondary 60J75, 31C25
- DOI: https://doi.org/10.1090/tran/6319
- MathSciNet review: 3413859