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arXiv · 0909.2301

Gibbs-like measure for spectrum of a class of one-dimensional Schrödinger operator with Sturm potentials

Shen Fan·Qing-Hui Liu·Zhi-Ying Wen
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Abstract

Let $α\in(0,1)$ be an irrational, and $[0;a_1,a_2,...]$ the continued fraction expansion of $α$. Let $H_{α,V}$ be the one-dimensional Schrödinger operator with Sturm potential of frequency $α$. Suppose the potential strength $V$ is large enough and $(a_i)_{i\ge1}$ is bounded. We prove that the spectral generating bands possess properties of bounded distortion, bounded covariation and there exists Gibbs-like measure on the spectrum $σ(H_{α,V})$. As an application, we prove that $$\dim_H σ(H_{α,V})=s_*,\quad \bar{\dim}_B σ(H_{α,V})=s^*,$$ where $s_*$ and $s^*$ are lower and upper pre-dimensions.

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Shen Fan, Qing-Hui Liu, Zhi-Ying Wen. 2009-09-12. Gibbs-like measure for spectrum of a class of one-dimensional Schrödinger operator with Sturm potentials. https://arxiv.org/abs/0909.2301

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