数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (5): 1997-2004.doi: 10.1007/s10473-023-0504-x

• • 上一篇    下一篇

THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS*

Qinghui LIU1,†, Zhiyi Tang1,2   

  1. 1. School of Computer Science and Technology, Beijing Institute of Technology, Beijing 100081, China;
    2. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
  • 收稿日期:2022-03-10 修回日期:2023-04-08 发布日期:2023-10-25

THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS*

Qinghui LIU1,†, Zhiyi Tang1,2   

  1. 1. School of Computer Science and Technology, Beijing Institute of Technology, Beijing 100081, China;
    2. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
  • Received:2022-03-10 Revised:2023-04-08 Published:2023-10-25
  • Contact: †Qinghui LIU, E-mail: qhliu@bit.edu.cn
  • About author:Zhiyi Tang, E-mail: tangzhiyi@hbpu.edu.cn
  • Supported by:
    National Natural Science Foundation of China (11871098).

摘要: Let $\tau$ be a generalized Thue-Morse substitution on a two-letter alphabet $\{a, b\}$:$\tau(a)=a^mb^m$, $\tau(b)=b^ma^ms$ for the integer $m\ge 2$. Let $\xi$ be a sequence in $\{a, b\}^{\mathbb{Z}}$ that is generated by $\tau$. We study the one-dimensional Schrödinger operator $H_{m, \lambda}$ on $l^2(\mathbb{Z})$ with a potential given by $$v(n)=\lambda V_{\xi}(n), $$ where $\lambda>0$ is the coupling and $V_\xi(n)=1$ ($V_\xi(n)=-1$) if $\xi(n)=a$ ($\xi(n)=b$). Let $\Lambda_2=2$, and for $m>2$, let $\Lambda_m=m$ if $m\equiv0\mod 4$; let $\Lambda_m=m-3$ if $m\equiv1\mod 4$; let $\Lambda_m=m-2$ if $m\equiv2\mod 4$; let $\Lambda_m=m-1$ if $m\equiv3\mod 4$. We show that the Hausdorff dimension of the spectrum $\sigma(H_{m, \lambda})$ satisfies that $$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$$ It is interesting to see that $\dim_H \sigma(H_{m, \lambda})$ tends to $1$ as $m$ tends to infinity.

关键词: one-dimensional Schrödinger operator, generalized Thue-Morse sequence, Hausdorff dimension

Abstract: Let $\tau$ be a generalized Thue-Morse substitution on a two-letter alphabet $\{a, b\}$:$\tau(a)=a^mb^m$, $\tau(b)=b^ma^ms$ for the integer $m\ge 2$. Let $\xi$ be a sequence in $\{a, b\}^{\mathbb{Z}}$ that is generated by $\tau$. We study the one-dimensional Schrödinger operator $H_{m, \lambda}$ on $l^2(\mathbb{Z})$ with a potential given by $$v(n)=\lambda V_{\xi}(n), $$ where $\lambda>0$ is the coupling and $V_\xi(n)=1$ ($V_\xi(n)=-1$) if $\xi(n)=a$ ($\xi(n)=b$). Let $\Lambda_2=2$, and for $m>2$, let $\Lambda_m=m$ if $m\equiv0\mod 4$; let $\Lambda_m=m-3$ if $m\equiv1\mod 4$; let $\Lambda_m=m-2$ if $m\equiv2\mod 4$; let $\Lambda_m=m-1$ if $m\equiv3\mod 4$. We show that the Hausdorff dimension of the spectrum $\sigma(H_{m, \lambda})$ satisfies that $$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$$ It is interesting to see that $\dim_H \sigma(H_{m, \lambda})$ tends to $1$ as $m$ tends to infinity.

Key words: one-dimensional Schrödinger operator, generalized Thue-Morse sequence, Hausdorff dimension

中图分类号: 

  • 28A78