Level spacings and periodic orbits

Abstract

Starting from a semiclassical quantization condition based on the trace formula, we derive a periodic-orbit formula for the distribution of spacings of eigenvalues with k intermediate levels. Numerical tests verify the validity of this representation for the nearest-neighbor level spacing (k=0). In a second part, we present an asymptotic evaluation for large spacings, where consistency with random matrix theory is achieved for large k. We also discuss the relation with the method of Bogomolny and Keating [Phys. Rev. Lett. 77, 1472 (1996)] for two-point correlations.

The Wigner distribution and 2D classical maps

“…An analytical means of understanding the spacing statistics of quantum chaotic systems, based on semiclassical periodic orbit theory, has been attempted in Ref. [20]. However, due to the asymptotic techniques employed therein, the semiclassical formulas for the kth-nearest-neighbor spacing distributions P (S; k) are good approximations to the corresponding distributions obtained from RMT only for large values of S and k; for small k and in particular for k = 1 the obtained semiclassical formulas do not reproduce the well-known RMT results.…”