Gibbs-like measure for spectrum of a class of one-dimensional Schr\"odinger operator with Sturm potentials

TL;DR

This paper establishes Gibbs-like measures for the spectrum of certain one-dimensional Schrödinger operators with Sturm potentials, revealing detailed fractal dimension properties of their spectra.

Contribution

It proves the existence of Gibbs-like measures and bounded distortion properties for the spectrum of Schrödinger operators with Sturm potentials under specific conditions.

Findings

Spectral generating bands have bounded distortion and covariation.

Existence of Gibbs-like measure on the spectrum.

Exact formulas for Hausdorff and box-counting dimensions of the spectrum.

Abstract

Let $\alpha\in(0,1)$ be an irrational, and $[0;a_1,a_2,...]$ the continued fraction expansion of $\alpha$. Let $H_{\alpha,V}$ be the one-dimensional Schr\"odinger operator with Sturm potential of frequency $\alpha$. 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 $\sigma(H_{\alpha,V})$. As an application, we prove that $$\dim_H \sigma(H_{\alpha,V})=s_*,\quad \bar{\dim}_B \sigma(H_{\alpha,V})=s^*,$$ where $s_*$ and $s^*$ are lower and upper pre-dimensions.