Volume 15, Issue 3
Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

Eric Bourgain-Chang

Commun. Comput. Phys., 15 (2014), pp. 712-732.

Published online: 2014-03

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  • Abstract

In this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper's equation. This study is motivated by various conjectures on the spectral theory of these 'pseudo-random' models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

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@Article{CiCP-15-712, author = {}, title = {Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective}, journal = {Communications in Computational Physics}, year = {2014}, volume = {15}, number = {3}, pages = {712--732}, abstract = {

In this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper's equation. This study is motivated by various conjectures on the spectral theory of these 'pseudo-random' models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.120513.290813a}, url = {http://global-sci.org/intro/article_detail/cicp/7112.html} }
TY - JOUR T1 - Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective JO - Communications in Computational Physics VL - 3 SP - 712 EP - 732 PY - 2014 DA - 2014/03 SN - 15 DO - http://doi.org/10.4208/cicp.120513.290813a UR - https://global-sci.org/intro/article_detail/cicp/7112.html KW - AB -

In this paper we perform a numerical study of the spectra, eigenstates, and Lyapunov exponents of the skew-shift counterpart to Harper's equation. This study is motivated by various conjectures on the spectral theory of these 'pseudo-random' models, which are reviewed in detail in the initial sections of the paper. The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model. In particular our numerics establish a small upper bound on the gaps in the spectrum (conjectured to be absent).

Eric Bourgain-Chang. (2020). Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective. Communications in Computational Physics. 15 (3). 712-732. doi:10.4208/cicp.120513.290813a
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