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A Fredholm determinant for semiclassical quantization.

Predrag Cvitanovic1, Per E. Rosenqvist, Gabor Vattay

  • 1Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, DenmarkUnite de Mathematiques Pures et Appliquees, Ecole Normale Superieure de Lyon 46, Allee d'Italie, F-69364 Lyon, France.

Chaos (Woodbury, N.Y.)
|October 1, 1993
PubMed
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We introduce the quantum Fredholm determinant, a novel approximation for quantum systems. Numerical tests suggest it offers improved analyticity and convergence over existing methods for hyperbolic flows.

Area of Science:

  • Quantum mechanics
  • Dynamical systems theory
  • Mathematical physics

Background:

  • Quantum determinants are crucial for understanding quantum systems.
  • Gutzwiller-Voros zeta functions, derived from the Gutzwiller trace formula, are used for approximations.
  • Axiom A hyperbolic flows present unique challenges in analysis.

Purpose of the Study:

  • To introduce and investigate the quantum Fredholm determinant as a new approximation.
  • To numerically test the conjecture that quantum Fredholm determinants offer superior analyticity and convergence.
  • To compare the performance of quantum Fredholm determinants against Gutzwiller-Voros zeta functions.

Main Methods:

  • Numerical investigation of the 3-disk repeller.
  • Analysis of a normal-form model of a hyperbolic flow.

Related Experiment Videos

  • Testing with a model 2-dimensional map.

    • Main Results:

    • The quantum Fredholm determinant shows a larger domain of analyticity in numerical tests.
    • The quantum Fredholm determinant demonstrates better convergence properties compared to Gutzwiller-Voros zeta functions.
    • Numerical evidence supports the conjecture for the tested models.

    Conclusions:

    • The quantum Fredholm determinant is a promising new approximation for quantum systems.
    • This method potentially offers advantages over existing techniques for analyzing hyperbolic flows.
    • Further research is warranted to explore its full potential in quantum chaos and related fields.