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Quantization conditions in Bogomolny's transfer operator method.

Cheng-Hung Chang1

  • 1National Center for Theoretical Sciences, Physics Division, 101, Section 2 Kuang-Fu Road, Hsinchu 300, Taiwan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 7, 2003
PubMed
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Bogomolny

Area of Science:

  • Quantum chaos
  • Mathematical physics
  • Classical and quantum dynamics

Background:

  • The study of quantum chaos utilizes methods like Bogomolny's transfer operator, Gutzwiller's trace formula, and dynamical zeta functions.
  • These methods generalize the Einstein-Brillouin-Keller quantization rule from integrable to chaotic systems.
  • The Fredholm determinant of the transfer operator on a Poincaré section quantifies the quantum system's energy spectrum.

Purpose of the Study:

  • To present two factorization formulas relating different quantization conditions.
  • To explain why various classical quantization conditions yield the same quantum energy spectrum.

Main Methods:

  • Application of Bogomolny's transfer operator method.
  • Derivation of factorization formulas for quantization conditions.

Related Experiment Videos

  • Illustration using the equilateral triangular billiard model.

    • Main Results:

    • Two novel factorization formulas are derived.
    • These formulas explicitly connect quantization conditions across different classical trajectory segments.
    • The equivalence of different classical quantization conditions for determining quantum energy spectra is demonstrated.

    Conclusions:

    • The derived factorization formulas provide a theoretical basis for the consistency of different quantization conditions in quantum chaos.
    • The findings clarify the relationship between classical dynamics and quantum energy spectra in chaotic systems.
    • The equilateral triangular billiard serves as a concrete example validating the theoretical results.