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Periodic orbit theory and spectral rigidity in pseudointegrable systems
Jesper Mellenthin1, Stefanie Russ
1Institut für Theoretische Physik III, Universität Giessen, D-35392 Giessen, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
Summary
This study numerically investigates pseudointegrable systems, finding their spectral rigidity aligns with quantum mechanics predictions. The shape of system corners influences spectral statistics, approaching integrable systems when corners are small.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Dynamical Systems
Background:
- Pseudointegrable systems, characterized by low genus numbers (g), present complex dynamics.
- Understanding spectral rigidity (Delta3(L)) is crucial for characterizing quantum systems.
- Rectangular systems with salient corners serve as models for pseudointegrable systems.
Purpose of the Study:
- To numerically calculate periodic orbits in pseudointegrable systems derived from rectangular shapes.
- To compute spectral rigidity (Delta3(L)) using semiclassical quantum mechanics from these orbits.
- To analyze the influence of system geometry, specifically salient corners, on spectral statistics.
Main Methods:
- Numerical calculation of periodic orbits for pseudointegrable systems with varying salient corner sizes.
- Application of semiclassical quantum mechanics to derive spectral rigidity (Delta3(L)) from periodic orbits.
- Comparison of spectral statistics derived from orbits with direct eigenvalue calculations.
Main Results:
- The diagonal approximation is validated for spectral rigidity calculations when averaged over energy intervals.
- Calculated spectral rigidity (Delta3(L)) shows good agreement with results from direct eigenvalue spectral statistics.
- System geometry significantly impacts spectral statistics: small corners lead to integrable-like behavior, while large corners create distinct pseudointegrable characteristics.
Conclusions:
- Periodic orbit properties effectively explain the observed spectral statistics in pseudointegrable systems.
- Diffraction terms appear to be minor contributors to spectral rigidity in the studied systems.
- The geometric features, particularly the size of salient corners, are key determinants of the transition between integrable and pseudointegrable spectral behaviors.