Random–Matrix Ensembles for Semi–Separable Systems

– 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 N → ∞) 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. It is usually assumed that the spectral fluctuation properties of classically chaotic systems coincide with those of the corresponding canonical ensemble of random–matrix theory (“quantum chaos conjecture”). There is abundant numerical evidence for the conjecture. In addition, several approaches have aimed at an analytical proof for the conjecture, see Section 5.9 of the review [1]. Naturally, these approaches have addressed generic systems. However, many of the commonly studied chaotic systems have a very particular form: They can be divided into two parts each of which is integrable, albeit in different coordinates. Such systems have been called semi–separable [2]. Typical cases are: The quarter stadium, kicked systems such as the kicked rotor, chaotic Jung scattering maps for integrable Hamiltonians [3], or Lombardi’s approximation for Rydberg molecules [4]. The spectral fluctuations of such semi–separable systems have been found to be essentially consistent with the quantum chaos conjecture. In view of the special nature of these systems, that finding is somewhat surprising. In the present letter, we construct two random–matrix models (one for the unitary and one for the orthogonal case) for semi–separable systems which we can solve analytically. The models take account of very specific features of semi–separable systems which are not reproduced by the circular ensembles of random–matrix theory. With the help of these models, we show why semi–separable systems nearly follow the predictions of standard random–matrix theory, and we predict quantitatively the deviations that are typically expected for such systems. In order