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Abstract 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.
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Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations
Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $$d$$ perturbations $$A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$$, $${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$$, with $${\operatorname{Ran}}{\textbf{B}}$$ being cyclic for $$A$$, parametrized by $$d\times d$$ Hermitian matrices $${\boldsymbol{\alpha }}$$, the singular parts of the spectral measures of $$A$$ and $$A_{\boldsymbol{\alpha }}$$ are mutually singular for all $${\boldsymbol{\alpha }}$$ except for a small exceptional set $$E$$. It was shown earlier by the 1st two authors, see [4], that $$E$$ is a subset of measure zero of the space $$\textbf{H}(d)$$ of $$d\times d$$ Hermitian matrices. In this paper, we show that the set $$E$$ has small Hausdorff dimension, $$\dim E \le \dim \textbf{H}(d)-1 = d^2-1$$.
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- Award ID(s):
- 1856719
- PAR ID:
- 10244170
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa281
