Sensitivity at the degenerate points of energy levels in a quantum system with nonintegrable perturbation

The behavior of quantum states and observables of a quasi-integrable quantum system is studied. Sensitivity of the quantum states and nonperiodicity of the average value of observables at the degenerate points of energy levels are observed. The behavior demonstrates the quantum properties corresponding to the classical resonant tori with rational frequency ratio, where the global structure is easily destroyed by a small perturbation. Our result is helpful for further understanding the avoided crossing in hard chaotic quantum systems.