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Quantization of linear maps on a torus-Fresnel diffraction by a periodic grating. (English) Zbl 1194.81107
Summary: Quantization on a phase space \(q, p\) in the form of a torus (or periodized plane) with dimensions \(\Delta q, \Delta p\) requires the Planck’s constant take one of the values \(h = \Delta q\Delta p/N\), where \(N\) is an integer. Corresponding to a linear classical map \(T\) of points \(q, p\) is a unitary operator \(U\) mapping quantum states that are periodic in \(q\) and \(p\); the construction of \(U\) involves techniques from number theory. \(U\) has eigenvalues \(\exp(i\alpha )\). The ‘eigenangles’ \(\alpha \) must be multiples of \(2\pi /n(N)\), where \(n(N)\) is the lowest common multiple of the lengths of the classical ‘cycles’ mapped under \(T\) by those rational points in q, p which are multiples of \(\Delta q/N\) and \(\Delta p/N\) (i.e. \(n(N)\) is the ‘period of \(T\) mod \(N^{\prime}\)), at least for odd \(N\). If \(T\) is hyperbolic, \(n\) is a very erratic function of \(N\), and the classical limit \(N \rightarrow \infty \) is very different from the ‘Bohr-Sommerfeld’ behaviour for parabolic maps. The degeneracy structure of the eigenangle spectrum is related to the distribution of cycle lengths. Computation of the quantal Wigner function shows that eigenstates of \(U\) do not correspond to individual cycles.

MSC:
81Q50 Quantum chaos
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