Dynamical symmetry of chaotic systems

The spectral fluctuation properties of a quantum system are related closely to the dynamical symmetries of the system. The equipotential lines of the system intuitively give information about the symmetry of the system, indicating the degree of symmetry breaking as well as how symmetry is lost. Chaotic motion appears when the curvature of the equipotential lines becomes negative. The energy spectrum of a helium atom is calculated, and by analyzing spectral fluctuations, it is found that the electron-electron interaction is not sufficient to change the integrable dynamical behavior of the two electrons in the central field of a nucleus. Even when this interaction is artificially amplified to the greatest possible extent below the dissociation limit, the Kolmogrov-Arnold-Moser theorem still qualitatively governs the quantum system. This coincides with the indication of the curvature of the equipotential lines that we discussed in this paper.