Level Clustering in the Regular Spectrum

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