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For systems whose classical dynamics is chaotic, it is generally believedthat the local statistical properties of the quantum energy levels are welldescribed by Random Matrix Theory. We present here two counterexamples - thehydrogen atom in a magnetic field and the quartic oscillator - which displaynearest neighbor statistics strongly different from the usual Wignerdistribution. We interpret the results with a simple model using a set ofregular states coupled to a set of chaotic states modeled by a random matrix.Comment: 10 pages, Revtex 3.0 + 4 .ps figures tar-compressed using uufiles  package, use csh to unpack (on Unix machine), to be published in Phys. Rev.  Let

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