Continuity of the measure of the spectrum for discrete quasiperiodic operators
Continuity of the measure of the spectrum for discrete quasiperiodic operators
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DOI：
10.4310/mrl.2002.v9.n4.a1
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发表时间：
2001-07
影响因子：
1
通讯作者：
S. Jitomirskaya;Igor Krasovsky
中科院分区：
文献类型：
--
作者：
S. Jitomirskaya;Igor Krasovsky
研究了l2（Z）上的离散Schrodinger算子（H ′，θ）（n）= n（n − 1）+n（n +1）+f（α n + θ）n（n），其中f（x）是周期为1的真实的解析周期函数.证明了在H ∈，∈的李雅普诺夫指数为正的条件下，H ∈，∈的谱的测度与其典型有理逼近的谱的测度之间的一个一般定理.对于几乎Mathieu算子（f（x）= 2 λ cos 2 π x），可以得出谱的测度等于4| 1 −| λ||对所有真实的θ，λ 6 ± 1和所有无理α。
We study discrete Schrodinger operators (H�,�ψ)(n) = ψ(n − 1) + ψ(n + 1) + f(αn+θ)ψ(n) on l 2 (Z), where f(x) is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of H�,� to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of H�,� are positive. For the almost Mathieu operator (f(x) = 2λcos2πx) it follows that the measure of the spectrum is equal to 4|1 − |λ|| for all real θ, λ 6 ±1, and all irrational α.