Spectral spacing correlations for Chaotic and Disordered systems

O. BOHIGAS, P.LEBOEUF, M.J.SÁNCHEZ

FOUNDATIONS OF PHYSICS

Año: 2001 vol. 31 p. 489 - 517

0015-9018

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point cor- relation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to dif- fusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron Frobenius operator, is derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear repli- cated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing ``data.´´ The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros.