Applications of periodic-orbit theory

Area of Science:

• Quantum chaos
• Mathematical physics
• Dynamical systems

Background:

• Gutzwiller's periodic-orbit theory connects classical periodic orbits to quantum mechanics.
• Understanding chaotic systems is a key challenge in physics.

Purpose of the Study:

• To apply Gutzwiller's theory using generalized periodic-orbit sum rules.
• To determine quantum energies efficiently in chaotic systems.

Main Methods:

• Numerical evaluation for the hyperbola billiard system.
• Formulation of a quantization condition using a zeta function and functional equation.

Main Results:

• Efficient semiclassical determination of quantum energies.
• Demonstration of the applicability of the zeta function quantization condition.

Conclusions:

• The generalized periodic-orbit sum rules provide an effective method for quantum energy determination.
• The zeta function approach offers an efficient route to quantizing chaotic systems.