Criticality for schrödinger type operators based on recurrent symmetric stable processes

Masayoshi Takeda

doi:10.1090/tran/6319

Criticality for schrödinger type operators based on recurrent symmetric stable processes

Masayoshi Takeda

SCI - Mathematics

Research output: Contribution to journal › Article

2 Citations (Scopus)

Abstract

Let μ be a signed Radon measure on ℝ¹ in the Kato class andconsider a Schrödinger type operator Hμ = (−d2/dx2)α/2 + μ on R1. Let 1 ≤ α < 2 and suppose the support of μ is compact. We then construct a bounded Hμ-harmonic function uniformly lower-bounded by a positive constant if Hμ is critical. Moreover, we show that there exists no bounded positive Hμ-harmonic function if Hμ is subcritical.

Original language	English
Pages (from-to)	149-167
Number of pages	19
Journal	Transactions of the American Mathematical Society
Volume	368
Issue number	1
DOIs	https://doi.org/10.1090/tran/6319
Publication status	Published - 2016 Jan

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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abstract = "Let μ be a signed Radon measure on ℝ1 in the Kato class andconsider a Schr{\"o}dinger type operator Hμ = (−d2/dx2)α/2 + μ on R1. Let 1 ≤ α < 2 and suppose the support of μ is compact. We then construct a bounded Hμ-harmonic function uniformly lower-bounded by a positive constant if Hμ is critical. Moreover, we show that there exists no bounded positive Hμ-harmonic function if Hμ is subcritical.",

author = "Masayoshi Takeda",

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AB - Let μ be a signed Radon measure on ℝ1 in the Kato class andconsider a Schrödinger type operator Hμ = (−d2/dx2)α/2 + μ on R1. Let 1 ≤ α < 2 and suppose the support of μ is compact. We then construct a bounded Hμ-harmonic function uniformly lower-bounded by a positive constant if Hμ is critical. Moreover, we show that there exists no bounded positive Hμ-harmonic function if Hμ is subcritical.

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