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

Published online by Cambridge University Press:  03 June 2015

Eric Bourgain-Chang
Eric Bourgain-Chang*
Affiliation:
Mechanical Engineering Department, University of California, Berkeley, CA 94720, USA
*
*Corresponding author.Email:ebc@berkeley.edu
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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).

Keywords

81Q1039A7047B39Schrödinger operatorskew-shiftspectrumlocalization

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Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 3 , March 2014, pp. 712 - 732
Copyright
Copyright © Global Science Press Limited 2014

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