Thin Spectra and Singular Continuous Spectral Measures for Limit-Periodic Jacobi Matrices

Abstract

This paper investigates the spectral properties of Jacobi matrices with limitperiodic coefficients. We show that for a residual set of such matrices, 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 can strengthen the result and show that the spectrum is a Cantor set of zero lower box counting dimension, and hence in particular of zero Hausdorff 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. Contents 1. Introduction and Results 1 2. Spectral Estimates for Periodic Jacobi Matrices 5 3. Singular Continuous Cantor Spectrum 12 Appendix A. The Gordon Lemma and Applications 14 Appendix B. The IDS for Periodic Jacobi Matrices 17 Acknowledgement 21 References 21 D.D.

Limit-Periodic Dirac Operators with Thin Spectra

“…Let us comment on the proofs of Theorems 1.1 and 1.2. Beginning with the seminal paper of Avila [1], there is by now a well-established path to obtaining thin spectra for limit-periodic operators, provided one can perform a version of Avila's perturb-and-grow technique [11,13,14,24]. The construction of [1] begins with a periodic operator, performs a finite number of small perturbations to move energies out of the spectrum, and exploits uniform hyperbolicity of cocycles in the resolvent set in conjunction with connections between the density of states and rotation number.…”

“…The first step is generally straightforward to implement, and follows readily from Floquet theory; compare [36]. The second step has traditionally relied on small translations (or dilations [13]) of the spectral measure in the self-adjoint setting, or rotations of the spectral measure in the unitary setting. In the Schrödinger setting, a translation of the spectral measure corresponds to a constant shift to the potential, so small translations correspond to uniformly small perturbations of the potential.…”