Avoided level crossings in an elliptic billiard.

Area of Science:

• Quantum mechanics
• Mathematical physics
• Classical and quantum chaos

Background:

• Elliptic billiards exhibit complex dynamics, including the localization of eigenfunctions on unstable periodic orbits.
• Understanding avoided level crossings is crucial for characterizing quantum systems with chaotic classical counterparts.

Purpose of the Study:

• To investigate the occurrence and nature of avoided level crossings in an elliptic billiard system.
• To demonstrate the sequential generation of eigenfunctions through these interactions.
• To identify the underlying resonance mechanism responsible for the observed phenomena.

Main Methods:

• Application of the Einstein-Brillouin-Keller quantization rule to unstable periodic orbits.
• Analysis of a perturbed Hamiltonian to study interactions between eigenfunctions.
• Verification of Fermi resonance by examining quantum number differences and periodic orbit localization.

Main Results:

• Observed Demkov-type avoided level crossings over wide ranges in the elliptic billiard.
• Demonstrated sequential generation of bouncing-ball modes localized on island chains and unstable periodic orbits.
• Confirmed non-zero off-diagonal Hamiltonian elements, indicating significant interaction between eigenfunctions.

Conclusions:

• The study confirms the existence of Demkov-type avoided level crossings in elliptic billiards.
• Fermi resonance is identified as the mechanism responsible for the generation and interaction of eigenfunctions.
• These findings provide insights into quantum chaos and spectral properties of complex systems.