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

^*Corresponding author for this work

School of Mathematics and Statistics

Tsinghua University

Research output: Contribution to journal › Article › peer-review

30 Citations (Scopus)

Abstract

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 < ∞.

Original language	English
Pages (from-to)	33-59
Number of pages	27
Journal	Potential Analysis
Volume	20
Issue number	1
DOIs	https://doi.org/10.1023/A:1025537823884
Publication status	Published - Feb 2004

Keywords

Hausdorff dimension
One-dimensional Schrödinger operators
Sturmian sequence

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abstract = "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 Schr{\"o}dinger 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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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 < ∞.

KW - Hausdorff dimension

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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Keywords

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