94-62 Degli Esposti M., Graffi S., Isola S.

Classical Limit of the Quantized Hyperbolic Toral Automorphisms (100K, AmS-TeX) Mar 10, 94

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Abstract. The canonical quantization of any hyperbolic symplectomorphism $A$ of the 2-torus yields a periodic unitary operator on a $N$-dimensional Hilbert space, $N=\frac1{h}$. We prove that this quantum system becomes ergodic and mixing at the classical limit ($N\to\infty $, $N$ 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 spread in phase space.

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