Statistical Properties of Energy Levels of Chaotic Systems: Wigner or Non-Wigner?

For systems whose classical dynamics is chaotic, it is generally believed that the local statistical properties of the quantum energy levels are well described by random matrix theory. We present here two counterexamples---the hydrogen atom in a magnetic field and the quartic oscillator---which display nearest neighbor statistics strongly different from the usual Wigner distribution. We interpret the results with a simple model using a set of regular states coupled to a set of chaotic states modeled by a random matrix.