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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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