Cantor Spectrum for the Almost Mathieu Operator
Cantor Spectrum for the Almost Mathieu Operator
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DOI：
10.1007/s00220-003-0977-3
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发表时间：
2004
影响因子：
2.4
通讯作者：
Joaquim Puig
中科院分区：
文献类型：
--
作者：
Joaquim Puig
在本文中，我们使用Almost Mathieu算子的可归约性、局部化和对偶性结果，$${{ {{\left({{H_{{b,\phi}} x}}\right)}}_n= x_{{n+1}} +x_{{n-1}} + b \cos{{\left({{2 \pi n \omega + \phi}}\right)}}x_n }}$$ onl2(ℤ) 及其关联的特征值方程推导出对于 b≠0、±2 和 ω 丢番图，算子的谱是实线的康托子集。这解决了带 ω 的这些值的所谓“十马提尼问题”。此外，我们证明，当 |b|≠0 足够小或足够大时，间隙标记定理预测的所有谱间隙都是开放的。
In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, $${{ {{\left({{H_{{b,\phi}} x}}\right)}}_n= x_{{n+1}} +x_{{n-1}} + b \cos{{\left({{2 \pi n \omega + \phi}}\right)}}x_n }}$$ onl2(ℤ) and its associated eigenvalue equation to deduce that forb≠0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ‘‘Ten Martini Problem’’ for these values ofband ω. Moreover, we prove that for |b|≠0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.