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

作　　者：刘庆晖 唐志毅 Qinghui LIU;Zhiyi TANG(School of Computer Science and Technology,Beijing Institute of Technology,Beijing 100081,China;School of Mathematics and Physics,Hubei Polytechnic University,Huangshi 435003,China)

机构地区：[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

出　　处：《Acta Mathematica Scientia》2023年第5期1997-2004,共8页数学物理学报（B辑英文版）

基　　金：supported by the National Natural ScienceFoundation of China(11871098)。

摘　　要：Letτbe a generalized Thue-Morse substitution on a two-letter alphabet{a,b}:τ(a)=ambm,τ(b)=bmam for the integer m≥2.Letξbe a sequence in{a,b}Z that is generated byτ.We study the one-dimensional Schr?dinger operator Hm,λon l2(Z)with a potential given by v(n)=λVξ(n),whereλ>0 is the coupling and Vξ(n)=1(Vξ(n)=-1)ifξ(n)=a(ξ(n)=b).LetΛ2=2,and for m>2,letΛm=m if m≡0 mod 4;letΛm=m-3 if m≡1 mod 4;letΛm=m-2if m≡2 mod 4;letΛm=m-1 if m≡3 mod 4.We show that the Hausdorff dimension of the spectrumσ(Hm,λ)satisfies that dimHσ(Hm,λ)>logΛm/(log 64m+4).It is interesting to see that dimHσσ(Hm,λ)tends to 1 as m tends to infinity.

关 键 词：one-dimensional Schrodinger operator generalized Thue-Morse sequence Hausdorff dimension 

分 类 号：O175[理学—数学]