The Hausdorff dimension of spectrum of a class of gerneralized Thue-Morse Hamiltonians

Abstract

We study a class of Schr\"odinger operators $H_{m,\lambda}$ with generalized Thue-Morse potential that generated by the substitution $\tau(a)=a^mb^m$, $\tau(b)=b^ma^m$ on two symbol alphabet $\Sigma=\{a,b\}$ for integer $m\ge 2$ and coupling $\lambda>0$. We show that $$\dim_H \sigma(H_{m,\lambda})\ge \frac{\log \Lambda_m}{\log 64m+4},$$ where $\sigma(H_{m,\lambda})$ is the spectrum of $H_{m,\lambda}$, $\Lambda_2=2$, and for $m>2$, $\Lambda_m=m$, if $m\equiv0\mod 4$; $\Lambda_m=m-3$, if $m\equiv1\mod 4$; $\Lambda_m=m-2$, if $m\equiv2\mod 4$; $\Lambda_m=m-1$, if $m\equiv3\mod 4$. This implies that $\dim_H \sigma(H_{m,\lambda})$ tends to $1$ as $m$ tends to infinity.