Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators
Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators
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次移和准周期薛定谔算子的奇异态密度测量
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Zhenghe Zhang
中科院分区：
文献类型：
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作者：
A. Avila;D. Damanik;Zhenghe Zhang
Simon的子移位猜想指出，对于Verblunsky系数的每一个非周期极小子移位，相关测度的公共本质支撑度为零。在本文中，我们用离散薛定谔算子的形式和它的类似公式证明了这个猜想是不成立的。此外，我们还证明了该猜想在薛定谔集下的一个弱版本。也就是说，在对子移位的一些附加假设下，我们证明了态密度度量是奇异的，它是与算子族相关的一种自然度量，其拓扑支撑度等于谱。我们还考虑了具有连续采样函数的单频准周期薛定谔算符，证明了一般情况下，态密度度量也是奇异的。
Simon’s subshift conjecture states that for every aperiodic minimal subshift of Verblunsky coefficients, the common essential support of the associated measures has zero Lebesgue measure. We disprove this conjecture in this paper, both in the form stated and in the analogous formulation of it for discrete Schrödinger operators. In addition we prove a weak version of the conjecture in the Schrödinger setting. Namely, under some additional assumptions on the subshift, we show that the density of states measure, a natural measure associated with the operator family and whose topological support is equal to the spectrum, is singular. We also consider one-frequency quasi-periodic Schrödinger operators with continuous sampling functions and show that generically, the density of states measure is singular as well.