Limit-Periodic Dirac Operators with Thin Spectra
Limit-Periodic Dirac Operators with Thin Spectra
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DOI：
10.1016/j.jfa.2022.109711
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发表时间：
2022-03
影响因子：
1.7
通讯作者：
B. Eichinger;J. Fillman;E. Gwaltney;Milivoje Luki'c
中科院分区：
文献类型：
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作者：
B. Eichinger;J. Fillman;E. Gwaltney;Milivoje Luki'c
证明了极限周期Dirac算子一般具有零Lebesgue测度的谱，并且它们的稠密集具有零Hausdorff维数的谱.证明结合了阿维拉的想法，从薛定谔设置一个新的交换参数产生开放的频谱间隙。这克服了以前在文献中观察到的一个障碍;即，在薛定谔型设置中，谱测度的平移对应于算子数据的小L∞扰动，但这对狄拉克或CMV算子不是真的。新的论点更加独立于模型。为了证明这一点，我们还应用该参数来证明具有有限周期Verblunsky系数的CMV矩阵的通用零测度谱。
We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schrödinger setting with a new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, in Schrödinger-type settings, translation of the spectral measure corresponds to small L∞-perturbations of the operator data, but this is not true for Dirac or CMV operators. The new argument is much more model-independent. To demonstrate this, we also apply the argument to prove generic zero-measure spectrum for CMV matrices with limit-periodic Verblunsky coefficients.