Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Qing Hui Liu; Zhi Ying Wen

doi:10.1023/A:1025537823884

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Qing Hui Liu^*, Zhi Ying Wen

^*此作品的通讯作者

数学学院

Tsinghua University

科研成果: 期刊稿件 › 文章 › 同行评审

28 引用（Scopus）

摘要

Let β ∈ (0, 1) be an irrational, and [a₁, a ₂,...] be the continued fraction expansion of β. Let H _β be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V > 20, then the Hausdorff dimension of the spectrum σ (H_β) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim inf_k→∞ (a₁a₂ ⋯ a_k))^1/k < ∞.

源语言	英语
页（从-至）	33-59
页数	27
期刊	Potential Analysis
卷	20
期	1
DOI	https://doi.org/10.1023/A:1025537823884
出版状态	已出版 - 2月 2004

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title = "Hausdorff Dimension of Spectrum of One-Dimensional Schr{\"o}dinger Operator with Sturmian Potentials",

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keywords = "Hausdorff dimension, One-dimensional Schr{\"o}dinger operators, Sturmian sequence",

author = "Liu, {Qing Hui} and Wen, {Zhi Ying}",

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N2 - Let β ∈ (0, 1) be an irrational, and [a1, a 2,...] be the continued fraction expansion of β. Let H β be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V > 20, then the Hausdorff dimension of the spectrum σ (Hβ) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim infk→∞ (a1a2 ⋯ ak))1/k < ∞.

AB - Let β ∈ (0, 1) be an irrational, and [a1, a 2,...] be the continued fraction expansion of β. Let H β be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V > 20, then the Hausdorff dimension of the spectrum σ (Hβ) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim infk→∞ (a1a2 ⋯ ak))1/k < ∞.

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KW - One-dimensional Schrödinger operators

KW - Sturmian sequence

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Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

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