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


Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective

Eric Bourgain-Chang 1*

1 Mechanical Engineering Department, University of California, Berkeley, CA 94720, USA.

Received 12 May 2013; Accepted (in revised version) 29 August 2013
Available online 15 November 2013
doi:10.4208/cicp.120513.290813a

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).

AMS subject classifications: 81Q10, 39A70, 47B39

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Key words: Schrodinger operator, skew-shift, spectrum, localization.

*Corresponding author.
Email: ebc@berkeley.edu
 

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