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The Arnol’d cat: Failure of the correspondence principle. (English) Zbl 0742.58024

The paper is devoted to a detailed investigation of the classical and quantal behaviour of a “kicked oscillator” system described by the Hamiltonian \[ H=p^ 2/2m + \varepsilon q^ 2/2\sum\delta (s-t/T) \] where the position \(q\) and the momentum \(p\) are Cartesian coordinates on the plane. The equations of motion for such Hamiltonian can be reduced to the integrable mapping of the plane \((q_{n+1},p_{n+1})=(q_ n+p_ n,q_ n+2p_ n)\). Under the periodic boundary conditions these equations become the Arnol’d cat map \((q_{n+1},p_{m+1})=(q_ n+p_ n,q_ n+2p_ n) (\mod 1)\). In the paper the Arnol’d cat is quantized and its quantum motion is examined for signs of chaos using algorithmic complexity as the litmus. The analysis reveals that the quantum cat is not chaotic in the deep quantum domain nor does it become chaotic in the classical limit as required by the correspondence principle. The authors conclude that the correspondence principle fails for the quantum Arnol’d cat.

MSC:

53D50 Geometric quantization
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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