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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Sturmian and spectral theory for discrete symplectic systems
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by Martin Bohner, Ondřej Došlý and Werner Kratz PDF
Trans. Amer. Math. Soc. 361 (2009), 3109-3123 Request permission

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
  • MR Author ID: 295863
  • ORCID: 0000-0001-8310-0266
  • Ondřej Došlý
  • 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
  • Received by editor(s): June 20, 2007
  • Published electronically: December 30, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 3109-3123
  • MSC (2000): Primary 39A12, 39A13, 34B24, 49K99
  • DOI: https://doi.org/10.1090/S0002-9947-08-04692-8
  • MathSciNet review: 2485420