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

Abstract

References

Related Articles 15

Metrics

Comments

Recommended 10

doi:10.1007/s10473-023-0504-x

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

CLC Number:

28A78

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.

Trendmd

[1] Bellissard J, Iochum B, Scoppola E, Testard D. Spectral properties of one dimensional quasi-crystals. Commun Math Phys, 1989, 125(3): 527-543
[2] Bovier A, Ghez J. Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions. Commun Math Phys, 1993, 158(1): 45-66
[3] Damanik D, Embree M, Gorodetski A, Tcheremchantsev S. The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Comm Math Phys, 2008, 280(2): 499-516
[4] Damanik D, Gorodetski A. Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Commun Math Phys, 2011, 305(1): 221-277
[5] Damanik D, Lenz D. A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem. Duke Math J, 2006, 133(1): 95-123
[6] Lenz D. Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals. Commun Math Phys, 2002, 227(1): 119-130
[7] Liu Q H, Qu Y H. Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Annales Henri Poincaré, 2011, 12(1): 153-172
[8] Liu Q H, Qu Y H. Uniform convergence of Schrödinger cocycles over bounded Toeplitz subshift. Annales Henri Poincaré, 2012, 13(6): 1483-1500
[9] Liu Q H, Qu Y H. On the Hausdorff dimension of the spectrum of Thue-Morse Hamiltonian. Commun Math Phys, 2015, 338(2): 867-891
[10] Liu Q H, Qu Y H, Yao X. The spectrum of period-doubling Hamiltonian. Commun Math Phys, 2022, 394(3): 1039-1100
[11] Liu Q H, Peyrière J, Wen Z Y. Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials. C R Math, 2007, 345(12): 667-672
[12] Liu Q H, Tan B, Wen Z X, Wu J. Measure zero spectrum of a class of Schrödinger operators. J Stat Phys, 2002, 106(3/4): 681-691
[13] Liu Q H, Wen Z Y. Hausdorff dimension of spectrum of one-dimensional Schr?dinger operator with Sturmian potentials. Potential Analysis, 2004, 20(1): 33-59
[14] Kolar M, Ali M K. Generalized Thue-Morse chains and their physical properties. Physical Review B, 1991, 43(1): 1034-1047
[15] Süto A. Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci hamiltonian. J Stat Phys, 1989, 56(3/4): 525-531
[16] Toda M.Theory of Nonlinear Lattices. 2nd enlarged ed. Solid-State Sciences 20. Tokyo: Springer-Verlag, 1989

[1]	Yajuan ZHAO, Yongsheng LI, Wei YAN, Xiangqian YAN. ON THE DIMENSION OF THE DIVERGENCE SET OF THE OSTROVSKY EQUATION [J]. Acta mathematica scientia,Series B, 2022, 42(4): 1607-1620.
[2]	Jun WANG, Zhenlong CHEN. HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS [J]. Acta mathematica scientia,Series B, 2022, 42(2): 653-670.
[3]	Dan LI, Junfeng LI, Jie XIAO. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s( $\mathbb{R}^n$ ) [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1223-1249.
[4]	Jianfeng ZHOU, Zhong TAN. REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1333-1365.
[5]	Zhenliang ZHANG, Chunyun CAO. ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LÜROTH EXPANSION [J]. Acta mathematica scientia,Series B, 2016, 36(1): 257-264.
[6]	CHEN Zhen-Long. ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES [J]. Acta mathematica scientia,Series B, 2014, 34(1): 141-161.
[7]	HU Dong-Gang, HU Xue-Hai. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES [J]. Acta mathematica scientia,Series B, 2013, 33(4): 943-949.
[8]	CHEN Zhen-Long. FRACTAL PROPERTIES OF POLAR SETS OF RANDOM STRING PROCESSES [J]. Acta mathematica scientia,Series B, 2011, 31(3): 969-992.
[9]	CHEN Zhen-Long, LI Hui-Qiong. POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS [J]. Acta mathematica scientia,Series B, 2010, 30(3): 857-872.
[10]	Chen Zhenlong. INTERSECTIONS AND POLAR FUNCTIONS OF FRACTIONAL BROWNIAN SHEETS [J]. Acta mathematica scientia,Series B, 2008, 28(4): 779-796.
[11]	Gao Junyang; Qiao Jianyong. CONTINUITY AND HAUSDORFF DIMENSION OF JULIA SET CONCERNING YANG-LEE ZEROS [J]. Acta mathematica scientia,Series B, 2008, 28(3): 530-536.
[12]	Hu Dihe; Zhang Xiaomin. THE DIMENSION FOR RANDOM SUB-SELF-SIMILAR SET [J]. Acta mathematica scientia,Series B, 2007, 27(3): 561-573.
[13]	Hu Dihe; Zhang Xiaomin. THE RANDOM SHIFT SET AND RANDOM SUB-SELF-SIMILAR SET [J]. Acta mathematica scientia,Series B, 2007, 27(2): 267-273.
[14]	Zhang Xiaomin; Hu Dihe. THE DIMENSIONS OF THE RANGE OF RANDOM WALKS IN TIME-RANDOM ENVIRONMENTS [J]. Acta mathematica scientia,Series B, 2006, 26(4): 615-628.
[15]	ZHANG Xiao-Qun, LIU Pan-Yan. THE REGULARITY OF RANDOM GRAPH DIRECTED SELF-SIMILAR SETS [J]. Acta mathematica scientia,Series B, 2004, 24(3): 485-492.

Viewed

Full text

	From	local

	Times	3
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	59

	From	Others

	Times	59
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	Sitan Lin. OBSTACLE PROBLEMS ON $RCD(K, N)$ METRIC MEASURE SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1925 -1944 .
[2]	Yanyan CUI, Zunfeng LI, Yonghong XIE, Yuying QIAO. THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN $\mathbb{C}^2$ ^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1571 -1586 .
[3]	Wei LUO, Zhaoyang YIN. LOCAL BIFURCATION OF STEADY ALMOST PERIODIC WATER WAVES WITH CONSTANT VORTICITY^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1633 -1644 .
[4]	Jinxia CEN, Stanis law MIGÓRSKI, Emilio VILCHES, Shengda ZENG. TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1645 -1667 .
[5]	Yirang Yuan, Changfeng Li, Tongjun Sun, Qing Yang. A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2026 -2042 .
[6]	Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2089 -2107 .
[7]	Qinghua XU, Weikang FANG, Weiheng FENG, Taishun LIU. THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2075 -2088 .
[8]	Yazhou CHEN, Hakho HONG, Xiaoding SHI. THE ASYMPTOTIC STABILITY OF PHASE SEPARATION STATES FOR COMPRESSIBLE IMMISCIBLE TWO-PHASE FLOW IN 3D^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2133 -2158 .
[9]	Xia Tao, Ziqing Xie. THE UNIFORM CONVERGENCE OF A DG METHOD FOR A SINGULARLY PERTURBED VOLTERRA INTEGRO-DIFFERENTIAL EQUATION^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2159 -2178 .
[10]	Lishuang Peng, Yong Li. NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2179 -2203 .