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Semiclassical Poincare map for integrable systems.

Bent Lauritzen1

  • 1Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.

Chaos (Woodbury, N.Y.)
|July 1, 1992
PubMed
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Semiclassical methods reveal quantum energy levels in systems like the rectangular billiard. Zeroes of a functional determinant precisely predict these energy levels via EBK quantization.

Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Dynamical Systems

Background:

  • Integrable systems, such as the rectangular billiard, are fundamental in classical and quantum mechanics.
  • Understanding quantization conditions in these systems is crucial for theoretical physics.

Purpose of the Study:

  • To apply the semiclassical Poincare map to integrable systems.
  • To demonstrate that zeroes of the functional determinant yield EBK quantization.
  • To analyze the properties of the transfer operator in this context.

Main Methods:

  • Application of the semiclassical Poincare map.
  • Analysis of the functional determinant.
  • Investigation of the transfer operator and its properties.

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Main Results:

  • The semiclassical Poincare map successfully applied to the rectangular billiard.
  • Zeroes of the functional determinant were shown to correspond to EBK quantization.
  • The transfer operator was found to be explicitly unitary and finite.

Conclusions:

  • The study provides a direct link between semiclassical methods and quantum quantization rules.
  • The finite nature of the transfer operator offers computational advantages.
  • This approach offers a new perspective on the Euler product expansion over periodic orbits.