论文标题
一类Gerneralized Thue-Morse Hamiltonians的Hausdorff尺寸
The Hausdorff dimension of spectrum of a class of gerneralized Thue-Morse Hamiltonians
论文作者
论文摘要
我们研究一类Schrödinger运营商$ h_ {m，λ} $，由替代$τ（a）= a^mb^m $，$τ（b）= b^ma^m $在两个符号alphabet $σ= = \ = \ {a，b \ \} $ for inte $ for $ m m c的$ 2和cou cou cou 2 $ c. cou 2 $ cou的替代$τ（a）= a^mb^m $，$τ（a）= a^mb^m $，$ H_ {m，λ} $中。我们表明，$$ \dim_hσ（h_ {m，λ}）\ ge \ frac {\logλ_m} {\ log 64m+4}，$σ（h_ {m，λ}）$是$ h_ {m，λ} $，$λ} $，$λ__2$，$ c $λ_2 $λ_m= m $，如果$ m \ equiv0 \ mod 4 $; $λ_m= m-3 $，如果$ m \ equiv1 \ mod 4 $; $λ_m= m-2 $，如果$ M \ equiv2 \ mod 4 $; $λ_m= m-1 $，如果$ M \ equiv3 \ mod 4 $。这意味着$ \dim_hσ（h_ {m，λ}）$倾向于$ 1 $ as as $ m $倾向于无限。
We study a class of Schrödinger operators $H_{m,λ}$ with generalized Thue-Morse potential that generated by the substitution $τ(a)=a^mb^m$, $τ(b)=b^ma^m$ on two symbol alphabet $Σ=\{a,b\}$ for integer $m\ge 2$ and coupling $λ>0$. We show that $$\dim_H σ(H_{m,λ})\ge \frac{\log Λ_m}{\log 64m+4},$$ where $σ(H_{m,λ})$ is the spectrum of $H_{m,λ}$, $Λ_2=2$, and for $m>2$, $Λ_m=m$, if $m\equiv0\mod 4$; $Λ_m=m-3$, if $m\equiv1\mod 4$; $Λ_m=m-2$, if $m\equiv2\mod 4$; $Λ_m=m-1$, if $m\equiv3\mod 4$. This implies that $\dim_H σ(H_{m,λ})$ tends to $1$ as $m$ tends to infinity.
