Random matrix ensembles for semi-separable systems

¹ Physics Department, Faculty of Mathematics and Physics University of Ljubljana - Slovenia
² Centro de Ciencias Fisicas, UNAM, Cuernavaca and Centro Internacional de Ciencias - Cuernavaca, Mexico
³ Max Planck Institut für Kernphysik - Heidelberg, Germany

Received: 15 September 2000
Accepted: 9 April 2001

Abstract

Many models for chaotic systems consist of joining two integrable systems with incompatible constants of motion. The quantum counterparts of such models have a propagator which factorizes into two integrable parts. Each part can be diagonalized. The two eigenvector bases are related by an orthogonal (or unitary) transformation. We construct a random matrix ensemble that mimics this situation and consists of a product of a diagonal, an orthogonal, another diagonal and the transposed orthogonal matrix. The diagonal phases are chosen at random and the orthogonal matrix from Haar's measure. We derive asymptotic results (dimension ) using Wick contractions. A new approximation for the group integration yields the next order in 1/N. We obtain a finite correction to the circular orthogonal ensemble, important in the long-range part of spectral correlations.