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

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

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

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

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

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doi:10.1007/s10473-023-0504-x

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

Qinghui LIU^1,†, Zhiyi Tang^1,2

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

Qinghui LIU^1,†, Zhiyi Tang^1,2

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) = λ V_{ξ} (n),

$v(n)=\lambda V_{\xi}(n),$ where

λ > 0

$\lambda>0$ is the coupling and

V_{ξ} (n) = 1

$V_\xi(n)=1$ (

V_{ξ} (n) = - 1

$V_\xi(n)=-1$ ) if

ξ (n) = a

$\xi(n)=a$ (

ξ (n) = b

$\xi(n)=b$ ). Let

Λ_{2} = 2

$\Lambda_2=2$ , and for

m > 2

$m>2$ , let

Λ_{m} = m

$\Lambda_m=m$ if

m \equiv 0 \mod 4

$m\equiv0\mod 4$ ; let

Λ_{m} = m - 3

$\Lambda_m=m-3$ if

m \equiv 1 \mod 4

$m\equiv1\mod 4$ ; let

Λ_{m} = m - 2

$\Lambda_m=m-2$ if

m \equiv 2 \mod 4

$m\equiv2\mod 4$ ; let

Λ_{m} = m - 1

$\Lambda_m=m-1$ if

m \equiv 3 \mod 4

$m\equiv3\mod 4$ . We show that the Hausdorff dimension of the spectrum

σ (H_{m, λ})

$\sigma(H_{m, \lambda})$ satisfies that

\dim_{H} σ (H_{m, λ}) > \frac{\log Λ_{m}}{\log 64 m + 4} .

$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$ It is interesting to see that

\dim_{H} σ (H_{m, λ})

$\dim_H \sigma(H_{m, \lambda})$ tends to

1

$1$ as

m

$m$ 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) = λ V_{ξ} (n),

$v(n)=\lambda V_{\xi}(n),$ where

λ > 0

$\lambda>0$ is the coupling and

V_{ξ} (n) = 1

$V_\xi(n)=1$ (

V_{ξ} (n) = - 1

$V_\xi(n)=-1$ ) if

ξ (n) = a

$\xi(n)=a$ (

ξ (n) = b

$\xi(n)=b$ ). Let

Λ_{2} = 2

$\Lambda_2=2$ , and for

m > 2

$m>2$ , let

Λ_{m} = m

$\Lambda_m=m$ if

m \equiv 0 \mod 4

$m\equiv0\mod 4$ ; let

Λ_{m} = m - 3

$\Lambda_m=m-3$ if

m \equiv 1 \mod 4

$m\equiv1\mod 4$ ; let

Λ_{m} = m - 2

$\Lambda_m=m-2$ if

m \equiv 2 \mod 4

$m\equiv2\mod 4$ ; let

Λ_{m} = m - 1

$\Lambda_m=m-1$ if

m \equiv 3 \mod 4

$m\equiv3\mod 4$ . We show that the Hausdorff dimension of the spectrum

σ (H_{m, λ})

$\sigma(H_{m, \lambda})$ satisfies that

\dim_{H} σ (H_{m, λ}) > \frac{\log Λ_{m}}{\log 64 m + 4} .

$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$ It is interesting to see that

\dim_{H} σ (H_{m, λ})

$\dim_H \sigma(H_{m, \lambda})$ tends to

1

$1$ as

m

$m$ tends to infinity.

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

中图分类号:

28A78

Qinghui LIU, Zhiyi Tang. THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS^*[J]. 数学物理学报(英文版), 2023, 43(5): 1997-2004.

Qinghui LIU, Zhiyi Tang. THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1997-2004.

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