Communications in Mathematical Physics

, Volume 167, Issue 3, pp 471–507 | Cite as

Classical limit of the quantized hyperbolic toral automorphisms

  • Mirko Degli Esposti
  • Sandro Graffi
  • Stefano Isola


The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N→∞,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.


Neural Network Statistical Physic Hilbert Space Phase Space Complex System 
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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Mirko Degli Esposti
    • 1
  • Sandro Graffi
    • 1
  • Stefano Isola
    • 1
  1. 1.Departmento di MatematicaUniversità di BolognaBolognaItaly

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