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Related Experiment Videos

Quantum zeta function for perturbed cat maps.

Stephen C. Creagh1

  • 1Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark.

Chaos (Woodbury, N.Y.)
|June 1, 1995
PubMed
Summary
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Semiclassical approximations for quantum cat maps remain accurate for eigenvalues, even when classical systems become nonhyperbolic. However, errors emerge in traces and characteristic polynomials as nonhyperbolic structures appear.

Area of Science:

  • Quantum mechanics
  • Classical dynamics
  • Mathematical physics

Background:

  • Semiclassical approximations are crucial for bridging quantum and classical mechanics.
  • Quantum cat maps provide a model system for studying quantum chaos.
  • Understanding the transition to nonhyperbolic regimes is key in dynamical systems.

Purpose of the Study:

  • To investigate the accuracy of semiclassical approximations for quantum cat maps.
  • To analyze the impact of classical nonhyperbolicity on these approximations.
  • To determine the validity of semiclassical methods under perturbation.

Main Methods:

  • Examining semiclassical approximations to quantum cat map spectra.
  • Perturbing the system to induce a transition to the nonhyperbolic regime.

Related Experiment Videos

  • Comparing approximation results with exact spectral properties.

    • Main Results:

    • Semiclassical approximations show initial accuracy for quantum cat map spectra.
    • Large errors appear in traces and characteristic polynomial coefficients post-nonhyperbolicity.
    • Eigenvalues derived from approximations maintain accuracy despite significant perturbations.

    Conclusions:

    • Semiclassical approximations remain surprisingly robust for eigenvalues in perturbed quantum cat maps.
    • The onset of nonhyperbolicity significantly affects certain spectral statistics.
    • Further research is needed to understand the limitations and applicability of these approximations.