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Tridiagonal Substitution Hamiltonians

Published online by Cambridge University Press:  17 July 2014

M. Mei  and
W. Yessen
M. Mei*
Affiliation:
Mathematics & Computer Science, Denison University, Granville, OH 43023-0810
W. Yessen*
Affiliation:
Mathematics, Rice University, 1600 Main St. MS-136, Houston, TX 77005
*
⋆Supported by the Michele T. Myers PD Account through Denison University. Part of thework presented herein was supported by DMS-0901627 (PI: A. Gorodetski)
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Abstract

We consider a family of discrete Jacobi operators on the one-dimensional integer latticewith Laplacian and potential terms modulated by a primitive invertible two-lettersubstitution. We investigate the spectrum and the spectral type, the fractal structure andfractal dimensions of the spectrum, exact dimensionality of the integrated density ofstates, and the gap structure. We present a review of previous results, some applications,and open problems. Our investigation is based largely on the dynamics of trace maps. Thiswork is an extension of similar results on Schrödinger operators, although some of theresults that we obtain differ qualitatively and quantitatively from those for theSchrödinger operators. The nontrivialities of this extension lie in the dynamics of theassociated trace map as one attempts to extend the trace map formalism from theSchrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are,in a sense, a test item, as many other models can be attacked via the same techniques, andwe present an extensive discussion on this.

Keywords

substitution Hamiltoniantrace map
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 204 - 238
Copyright
© EDP Sciences, 2014

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