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Differentiability of spectral functions for symmetric $ \alpha$-stable processes


Authors: Masayoshi Takeda and Kaneharu Tsuchida
Journal: Trans. Amer. Math. Soc. 359 (2007), 4031-4054
MSC (2000): Primary 60J45, 60J40, 35J10
DOI: https://doi.org/10.1090/S0002-9947-07-04149-9
Published electronically: March 20, 2007
MathSciNet review: 2302522
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Abstract: Let $ \mu$ be a signed Radon measure in the Kato class and define a Schrödinger type operator $ \mathcal{H}^{\lambda\mu}=\frac{1}{2}(-\Delta)^{\frac{\alpha}{2}} + \lambda\mu$ on $ \mathbb{R}^d$. We show that its spectral bound $ C(\lambda)=-\inf\sigma(\mathcal{H}^{\lambda\mu})$ is differentiable if $ \alpha<d\leq 2\alpha$ and $ \mu$ is Green-tight.


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Additional Information

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

Kaneharu Tsuchida
Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan
Email: kanedon@ma8.seikyou.ne.jp

DOI: https://doi.org/10.1090/S0002-9947-07-04149-9
Keywords: Symmetric stable process, spectral function, criticality, additive functional, Kato measure
Received by editor(s): February 25, 2004
Received by editor(s) in revised form: August 16, 2005
Published electronically: March 20, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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