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

Abstract

We study a class of Schrödinger operators $H_{m,\lambda }$ with generalized Thue-Morse potential that generated by the substitution $\tau (a)=a^m{b}^m$, $\tau (b)=b^m{a}^m$ 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 $m\equiv 0\mathrm{mod}4$; ${\Lambda }_m=m-3$, if $m\equiv 1\mathrm{mod}4$; ${\Lambda }_m=m-2$, if $m\equiv 2\mathrm{mod}4$; ${\Lambda }_m=m-1$, if $m\equiv 3\mathrm{mod}4$. This implies that ${\dim }_H\sigma (H_{m,\lambda })$ tends to $1$ as $m$ tends to infinity.

10 pages, 1 figure