Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices
Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices
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DOI：
10.1002/mana.202100561
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发表时间：
2023-06
影响因子：
1
通讯作者：
David Damanik;J. Fillman;Chunyi Wang
中科院分区：
文献类型：
--
作者：
David Damanik;J. Fillman;Chunyi Wang
本文研究了具有极限周期系数的Jacobi矩阵的谱性质。我们证明了一般谱是零Lebesgue测度的Cantor集，并且谱测度是纯奇异连续的。对于有限周期Jacobi矩阵的稠密集，我们证明了谱是一个下盒维数为零的Cantor集，同时仍然保持谱型的奇异连续性。我们还展示了如何通过固定非对角系数并仅改变对角系数来建立这种性质的结果，以及在更严格的版本中，通过固定对角系数为零并仅改变非对角系数。我们应用这些结果来产生具有纯奇异连续谱型和零维谱的多维整数格上的加权拉普拉斯算子的例子。
This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum.