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

Abstract

Let $\hat{I}\pm \in (0,1)$ be an irrational, and $[0;a_1,a_2,...]$ the continued fraction expansion of $\hat{I}\pm$. Let $H_{\hat{I}\pm ,V}$ be the one-dimensional Schrödinger operator with Sturm potential of frequency $\hat{I}\pm$. Suppose the potential strength $V$ is large enough and $(a_i{)}_{i\ge 1}$ 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 $\ddot{I}ƒ(H_{\hat{I}\pm ,V})$. As an application, we prove that $${\dim }_H\ddot{I}ƒ(H_{\hat{I}\pm ,V})=s_{\ast },{\bar{\dim }}_B\ddot{I}ƒ(H_{\hat{I}\pm ,V})=s^{\ast },$$ where $s_{\ast }$ and $s^{\ast }$ are lower and upper pre-dimensions.

19 pages