  Contrary to conventional wisdom, level repulsion in semiclassical spectrum is
not just a feature of classically chaotic systems, but classically integrable
systems as well. While in chaotic systems level repulsion develops on a scale
of the mean level spacing, regardless of location in the spectrum, in
integrable systems it develops on a much longer scale - geometric mean of the
mean level spacing and the running energy in the spectrum. We show that at this
scale level correlations in integrable systems have a universal dependence on
level separation, as well as discuss their exact form at any scale. These
correlations have dramatic consequences, including deviations from Poissonian
statistics in the nearest level spacing distribution and persistent
oscillations of level number variance as a function of the interval width. We
illustrate our findings on two models - a rectangular infinite well and a
modified Kepler problem - that serve as generic types of hard-wall billiards
and potential problems. Our theory is based on the concept of parametric
averaging, which allows for a statistical ensemble of integrable systems at a
given spectral location (running energy).
