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The quantized Baker’s transformation. (English) Zbl 0664.58045

A quantum analogue of the Baker’s transformation is constructed using a specially developed quantization procedure. We obtain a unitary operator acting on an N-dimensional Hilbert space, with N finite (and even), that has similar properties to the classical Baker’s map, and reduces to it in the classical limit, which corresponds here to \(N\to \infty\). The operator can be described as a very simple, fully explicit \(N\times N\) matrix. Numerical investigations confirm that this model has non-trivial features which ought to represent quantal manifestations of classical chaoticity. The quasi-energy spectrum is given by irrational eigenangles, leading to no recurrences. Most eigenfunctions look irregular, but some exhibit puzzling regular features, such as peaks at coordinate values belonging to periodic orbits of the classical Baker’s map. We compare the quantal and classical time-evolutions, as applied to initially coherent quasi-classical states: the evolving states stay in close agreement for short times but seem to lose all relationship to each other beyond a critical time of the order of \(\log_ 2N\sim -\log \hslash\).

MSC:

58Z05 Applications of global analysis to the sciences
53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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