Level clustering in the regular spectrum

In the regular spectrum of an $f$-dimensional system each energy level can be labelled with $f$ quantum numbers originating in $f$ constants of the classical motion. Levels with very different quantum numbers can have similar energies. We study the classical limit of the distribution $P(S)$ of spacings between adjacent levels, using a scaling transformation to remove the irrelevant effects of the varying local mean level density. For generic regular systems $P(S)$ = e$^{-S}$, characteristic of a Poisson process with levels distributed at random. But for systems of harmonic oscillators, which possess the non-generic property that the 'energy contours' in action space are flat, $P(S)$ does not exist if the oscillator frequencies are commensurable, and is peaked about a non-zero value of $S$ if the frequencies are incommensurable, indicating some regularity in the level distribution; the precise form of $P(S)$ depends on the arithmetic nature of the irrational frequency ratios. Numerical experiments on simple two-dimensional systems support these theoretical conclusions.