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The asymptotic-energy limit of the density P ( S ) of spacings between adjacent levels of the two-dimensional harmonic oscillator ( T D H O ) spectrum is studied. It is shown that in any integer segment [ M , M + 11, containing -h f / a levels, of the TDHO spectrum m + an, P ( S ) has the form 1 w , S ( S -S,) where i takes on at most three values. For large M , P ( S ) displays strong level repulsion for irrational a, but it does not settle on a stationary form nor does its average over M . This is in marked contrast with the behaviour of generic integrable systems for which the Poisson statistics, P ( S ) = exp(-S), is known to apply.

For a given integer d, 1 ≤ d ≤ n − 1, let Ω be a subset of the set of all d × n real matrices. Define the subspace M(Ω) = span{g(Ax) : A ∈ Ω, g ∈ C(IR d , IR)}. We give necessary and sufficient conditions on Ω so that M(Ω) is dense in C(IR n , IR) in the topology of uniform convergence on compact subsets. This generalizes work of Vostrecov and Kreines. We also consider some related problems.

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