Energy Level Quasi-Crossings: Accidental Degeneracies or Signature of Quantum Chaos?

In the field of quantum chaos, the study of energy levels plays an important role. The aim of this review paper is to critically discuss some of the main contributions regarding the connection between classical dynamics, semi-classical quantization and spectral statistics of energy levels. In particular, we analyze in detail degeneracies and quasi-crossings in the eigenvalues of quantum Hamiltonians which are classically non-integrable. Summary: 1. Introduction; 2. Quasi-Crossing and Chaos; 3. Molecular Spectroscopy; 4. Nuclear Models; 4.1 Zirnbauer-Verbaashot-Weidenmuller Model; 4.2 Lipkin-Meshow-Glick Model; 5. Particle Physics and Field Theory; 6. Conclusions.