ON THE PROPERTIES OF LEVEL SPACINGS FOR DECOMPOSABLE SYSTEMS

In this paper we show that the quantum theory of chaos, based on the s tatistical theory of energy spectra, presents inconsistencies difficul t to overcome. In classical mechanics a system described by a Hamilton ian H = H-1 + H-2 (decomposable) cannot be ergodic, because there are always two dependent integrals of motion besides the constant of energ y. In quantum mechanics we prove the existence of decomposable systems H-q = H-1(q) + H-2(q) whose spacing distribution agrees with the Wign er law and we show that in general the spacing distribution of H-q is not the Poisson law, even if it has often the same qualitative behavio ur. We have found that the spacings of H-q are among the solutions of a well-defined class of homogeneous linear systems. We have obtained a n explicit formula for the bases of the kernels of these systems, and a chain of inequalities which the coefficients of a generic linear com bination of the basis vectors must satisfy so that the elements of a p articular solution will be all positive, i.e. can be considered a set of spacings.