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Rotation Number and Exponential Dichotomy for Linear Hamiltonian Systems: From Theoretical to Numerical Results

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Abstract

The paper considers the rotation number for a family of linear nonautonomous Hamiltonian systems and its relation with the exponential dichotomy concept. We propose numerical techniques to compute the rotation number and we employ them to infer when a given system enjoys or not an exponential dichotomy. Comparisons with QR-based techniques for exponential dichotomy will give new insights on the structure of the spectrum for the one-dimensional quasi-periodic Schrödinger operator. Experiments on the two dimensional Schrödinger equation will be presented as well.

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Acknowledgments

The authors wish to thank Professors Luca Dieci and Russell Johnson for stimulating discussions, useful comments and suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bari, 70125 , Bari, Italy

    Cinzia Elia

  2. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università di Firenze, 50139 , Firenze, Italy

    Roberta Fabbri

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  1. Cinzia Elia
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  2. Roberta Fabbri
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Correspondence to Roberta Fabbri.

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Elia, C., Fabbri, R. Rotation Number and Exponential Dichotomy for Linear Hamiltonian Systems: From Theoretical to Numerical Results. J Dyn Diff Equat 25, 95–120 (2013). https://doi.org/10.1007/s10884-013-9290-9

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  • DOI: https://doi.org/10.1007/s10884-013-9290-9

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