On the Spectra of Separable 2D Almost Mathieu Operators
On the Spectra of Separable 2D Almost Mathieu Operators
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DOI：
10.1007/s00023-021-01080-x
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发表时间：
2020-12
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影响因子：
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通讯作者：
Alberto Takase
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文献类型：
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作者：
Alberto Takase
我们考虑由一维几乎Mathieu算子生成的可分二维离散薛定谔算子。对于固定的丢番图频率，我们证明了足够小的耦合的频谱必须是一个区间。这补充了J. Bourgain的一个结果，即对于固定耦合，频谱对于某些（正测量）丢番图频率具有间隙。我们的结果推广到可分离的多维离散薛定谔算子所产生的一维准周期算子，其潜力是解析的，其频率是丢番图。证明是基于几乎Mathieu算子的谱的厚度的研究，并利用纽豪斯间隙引理的康托集的总和。
We consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets.