替换系统的谱逼近
Spectral approximation for substitution systems
AI总结：
本文研究 Lie 群格点上替换系统势对应的非周期 Schrödinger 算子，利用有限图刻画谱的 Hausdorff 收敛性，并给出高维不可周期逼近及指数收敛速率的新结论。
摘要：
我们研究具有伸缩结构的 Lie 群上格点中非周期 Schrödinger 算子的周期逼近。其势由近期在此背景下引入的符号替换系统产生。我们通过有限图的性质，刻画了相关 Schrödinger 算子谱在 Hausdorff 距离下的收敛性。作为推论，得到了周期逼近的新例子。我们进一步证明，与一维情形不同，在高维中存在不允许周期逼近的替换系统。另一方面，我们表明，如果谱收敛，则收敛速率必然是指数级快的。即使对于 $\mathbb{Z}^d$ 上的替换，这些结果也是新的。
英文摘要：
We study periodic approximations of aperiodic Schrödinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schrödinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$.
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