An Interdisciplinary Journal

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

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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Last updated: *June 16, 2001*