具有不连续单调势的拟周期 Schrödinger 算子谱中间隙的研究
On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials
AI总结：
本文利用秩一扰动下 Anderson 局域化的稳定性，证明一维离散 Schrödinger 算子谱中无区间，并据此确立锯齿形势拟周期算子的 Cantor 谱，同时给出间隙填充结果。
摘要：
我们证明，对于一维离散 Schrödinger 算子，一类秩一扰动下 Anderson 局域化的稳定性意味着谱中不存在区间。该论证基于 Gordon 以及 del Rio–Makarov–Simon 的著名结果，并结合一种处理值域不一定是循环的扰动的方法。这些结果的主要应用是证明：一类具有锯齿形势的拟周期算子（已知其存在这种形式的稳定局域化）具有 Cantor 谱。对于某些一般的（不一定单调的）具有不连续势的拟周期算子类别，我们还得到了若干关于秩一扰动下间隙填充的结果。
英文摘要：
We show that, for one-dimensional discrete Schrödinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and del Rio--Makarov--Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.
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