Nonlinear Phenomena in Complex Systems, 2000, Vol.3, No.3, pp.260--267

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2000, Volume 3, Number 3, pp.260--267

Quantum Poincaré Mapping: Systems with Smooth Potentials.
Tomaz Prosen

Exact quantization of the Poincare mapping over the surface of section of an autonomous Hamiltonian system is shortly reviewed. The method reduces the number of freedoms in the system by one. Further we propose an efficient and stable numerical scheme for the computation of the unitary quantum Poincare mapping in the general case of smooth potentials. We illustrate the proposed method by working out the two examples: The diamagnetic Kepler problem (hydrogen atom in a uniform magnetic field) and the Nelson potential. It is demonstrated explicitly that the results of the method (say the system’s energy spectrum) do not depend on the choice of surface of section. The latter may even lie in the classically forbidden part of phase space.
Key words: quantum Poincare mapping, quantization, diamagnetic Kepler problem representation of supersymmetry.

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