Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits |
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| Authors: | Cheng-Hung Chang Tyll Krüger Roman Schubert Serge Troubetzkoy |
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| Institution: | 1.Institute of Physics,National Chiao Tung University,Hsinchu,Taiwan, ROC;2.Taiwan National Center for Theoretical Sciences,Hsinchu,Taiwan, ROC;3.Technische Universit?t Berlin,Institut für Mathematik,Berlin,Germany;4.School of Mathematics,University of Bristol, University Walk,Bristol,UK;5.Centre de physique théorique, Fédération de Recherches des Unités de Mathématiques de Marseille,Institut de mathématiques de Luminy and Universitée la Méditerranée,Marseille Cedex 9,France |
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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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