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首页> 外文期刊>Annales Henri Poincare >Continuum Schr?dinger Operators Associated With Aperiodic Subshifts
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Continuum Schr?dinger Operators Associated With Aperiodic Subshifts

机译：与非周期性子移位相关的连续薛定ding算子

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We study Schr?dinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik- Lenz and Klassert-Lenz-Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible.We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke-Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.
机译：我们在实线上研究Schrdinger算子，其势是由有限的字母上的基础遍历子移位和通过紧密支持的势块替换符号的规则所产生的。我们首先发展标准理论，该理论表明光谱和光谱类型几乎可以肯定是恒定的，并且可以通过随着Lyapunov指数的消失基本关闭能量来确定几乎确定的绝对连续光谱。使用Damanik-Lenz和Klassert-Lenz-Stollmann的结果，我们还表明，如果子位移满足Boshernitzan条件且电势是非周期性且不可约的，则频谱是零勒贝格测度的Cantor集。根据相关的Fricke-Vogt不变量的值，对子位移进行了详细的子移位，并证明了在给定能量下频谱的局部Hausdorff维数的结果。这些结果说明了局部势能块的一些简单选择，例如分段常数和局部点相互作用。在后一种特殊情况下，我们的结果解释了在物理学文献中已指出的所谓伪带的出现。

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