Level-spacing distribution of the harmonic oscillator.

Abstract

In this paper the nearest-neighbor level-spacing distributions P(s) of the two-dimensional harmonic oscillator for which the energy contours are flat in the action space are calculated. It is found that P(s) for a given level group on an energy contour is not peaked about a nonzero value of s regardless of whether the oscillator frequency ratios are irrational or rational. The precise form of P(s) is a \ensuremath{\delta} function independent of the arithmetic nature of the frequency ratios. We propose another method of constructing P(s) in which many sets of levels are taken as a whole. Using this method, some surprising distributions, such as Gaussian orthogonal ensemble-like, Poisson-like, and unit-step-function-like distributions, are exhibited for some sets of levels far away from the scaled energy contours of the harmonic oscillator.