Level Clustering

Abstract

Here I propose to consider classical autonomous systems with f degrees of freedom and 2f pairs of canonical variables p _i , q _i . We shall meet with invariant tori, caustics, and Maslov indices and proceed to the semiclassical torus quantization à la Einstein, Brillouin and Keller (EBK) and its modern variant, a periodicorbit theory. The latter will allow us to understand why there is no repulsion but rather clustering of levels for generic integrable systems with two or more degrees of freedom; the density of level spacings therefore usually takes the form of a single exponential, P(S) = e^−S. Single-freedom systems, if autonomous, are always integrable and do not respect any general rule for their spacing statistics; we shall postpone a discussion of their behavior to the subsequent chapter.