论文标题
极限周期性狄拉克操作员薄光谱
Limit-Periodic Dirac Operators with Thin Spectra
论文作者
论文摘要
我们证明，极限 - 周期性狄拉克运算符通常具有零lebesgue度量的光谱，并且一组密集的光谱具有零Hausdorff尺寸的光谱。证明将阿维拉（Avila）的思想与施罗德（Schrödinger）环境的想法结合在一起，并为产生开放频谱差距的新换向论点。这克服了先前在文献中观察到的障碍。也就是说，在Schrödinger-type设置中，光谱度量的翻译对应于操作员数据的小$ l^\ infty $ erpertations，但对于DIRAC或CMV操作员来说并不是这样。新的论点更独立于模型。为了证明这一点，我们还使用该参数来证明具有极限周期性Verblunsky系数的CMV矩阵的通用零量谱。
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 $L^\infty$-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.
