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.
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
.
DOI: 10.1007/s10473-023-0504-x
[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
