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Kuznetsov S. P. Arnold’s cat map: quantum chaos and operator dynamics in Heisenberg representation. Izvestiya VUZ. Applied Nonlinear Dynamics, 1998, vol. 6, iss. 3, pp. 3-48.

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Russian
Heading: 
Quantum Chaos
Article type: 
Article
UDC: 
517.9

Arnold’s cat map: quantum chaos and operator dynamics in Heisenberg representation

Autors: 
Kuznetsov Sergey Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The quantum model system is considered for which the classic analog is the known Arnold’s саг map. Due to periodicity conditions for the phase space, quantum states are represented by vectors of finite dimension N, and operators by NxN matrices. The integer parameter N characterizes a relative value of quantum effects; classic limit corresponds to N—>∞. Operator тар is suggested which governs discrete time evolution in Heisenberg representation for operators of finite shifts for position and momentum. Explicit form of evolution operator is stated in Schrodinger representation. Solution for non—stationary problem is presented and discussed for initial conditions taken as localized state, two delta—spikes, Gaussian wave packet. Quantum dynamics in terms of Husimi distribution and Wigner function, quasi—cnergy spectrum and eigenvector structure are discussed on а basis of dynamics of Heisenberg operators.

Key words: 
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Acknowledgments: 
The work was supported by the RFBR (project N 97-02-16414).
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Received: 
10.04.1998
Accepted: 
20.10.1998
Published: 
15.01.1999
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