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Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits

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Abstract

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions.

We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits.

The maps we quantise are obtained by cutting and stacking.

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Authors and Affiliations

  1. Institute of Physics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan, ROC

    Cheng-Hung Chang

  2. Taiwan National Center for Theoretical Sciences, 101 Section 2 Kuang Fu Road, Hsinchu, 300, Taiwan, ROC

    Cheng-Hung Chang

  3. Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Tyll Krüger

  4. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Roman Schubert

  5. Centre de physique théorique, Fédération de Recherches des Unités de Mathématiques de Marseille, Institut de mathématiques de Luminy and Universitée la Méditerranée, Luminy, Case 907, F-13288, Marseille Cedex 9, France

    Serge Troubetzkoy

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  1. Cheng-Hung Chang
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  2. Tyll Krüger
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  3. Roman Schubert
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  4. Serge Troubetzkoy
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Communicated by P. Sarnak

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Chang, CH., Krüger, T., Schubert, R. et al. Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits. Commun. Math. Phys. 282, 395–418 (2008). https://doi.org/10.1007/s00220-008-0557-7

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  • DOI: https://doi.org/10.1007/s00220-008-0557-7

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