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Sturmian and spectral theory for discrete symplectic systems

Authors: Martin Bohner, Ondrej Dosly and Werner Kratz
Journal: Trans. Amer. Math. Soc. 361 (2009), 3109-3123
MSC (2000): Primary 39A12, 39A13, 34B24, 49K99
Published electronically: December 30, 2008
MathSciNet review: 2485420
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Abstract: We consider $ 2n\times 2n$ symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most $ n$. In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.

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Additional Information

Martin Bohner
Affiliation: Department of Mathematics and Statistics and Department of Economics and Finance, Missouri University of Science and Technology, Rolla, Missouri 65401

Ondrej Dosly
Affiliation: Department of Mathematics and Statistics, Masaryk University, CZ-61137, Brno, Czech Republic

Werner Kratz
Affiliation: Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany

Keywords: Discrete symplectic system, discrete quadratic functional, Sturmian separation result, Sturmian comparison result, Rayleigh principle, extended Picone identity
Received by editor(s): June 20, 2007
Published electronically: December 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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