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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Criticality for Schrödinger type operators based on recurrent symmetric stable processes
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by Masayoshi Takeda PDF
Trans. Amer. Math. Soc. 368 (2016), 149-167 Request permission

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