Classical limit of the quantized hyperbolic toral automorphisms. (English) Zbl 0822.58022
Summary: The canonical quantization of any hyperbolic symplectomorphism \(A\) of the 2-torus yields a periodic unitary operator on an \(N\)-dimensional Hilbert space, \(N = {1\over h}\). 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.
MSC:
| 53D50 | Geometric quantization |
| 81S20 | Stochastic quantization |
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
Keywords:
quantum ergodicity; quantization; hyperbolic symplectomorphism; 2-torus; mixing; stochastic behaviour; Wigner functions; eigenstates; phase spaceReferences:
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