Quantal energy spectra for a quartic potential

The classical Hamiltonian Hα(p, q) = 1/2(p12 + p22) + Uα(q) with 0 α 1 and Uα(q) = (1 – α)/12(q14 + q24) + 1/2q12q22 is integrable for α = 0. For α = 1 the motion is always irregular except for special orbits. Here we study the energy spectra of the quantized version and discuss the connection with "quantum chaos". For α = 0 the distribution of the nearest neighbour energy level spacings (in the invariant subspaces) is given by a Poisson distribution. With increasing α the distribution becomes more and more of Wigner type. For α = 1 we find a pure Wigner distribution. We also discuss the connection with a theorem due to von Neumann and Wigner. No energy level crossing occurs (in the invariant subspaces) with increasing α. We obtain level repulsion with increasing α. Extended Hamiltonians are briefly considered.

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