Statistics of energy spectra

When there is a stochastic disruption of the integrals of motion of a system, the corresponding quantum numbers disappear. The energy spectrum of the system becomes quasirandom. In the present paper, the quantization rules and the distribution of distances between pairs of adjacent levels are studied for this case. The probability for the appearance of very closely spaced levels is governed by a critical index which is expressed in terms of the Kolmogorov entropy for the given system (i.e., in terms of the growth rate for the instability of the classical trajectories in the corresponding phase space). Various physical situations are discussed in which a random spectral structure can arise. The capabilities of a quasiclassical analysis in the case of a stochastic disruption of the integrals of motion are discussed.