It is by now well established that spectral fluctuations of quantum systems whose underlying classical motion is chaotic exhibit universalities. On the other hand, it is also well known that there are parameters characterizing a particular chaotic system. The purpose of the two lectures will be to illustrate, partly by studying particular examples, what makes some properties system-dependent and other universal. The general framework will be periodic orbit and random matrix theories.
Emphasis will be put on two properties not so widely discussed so far: i) spectral spacing autocorrelations, ii) total energies of (non-interacting) fermion systems. Besides a general discussion, two cases will be treated in detail: 1) zeros of the Riemann zeta function (an example of a 'chaotic' system for which all 'classical' information is well known), 2) the fluctuation of the binding energy of atomic nuclei (a system which is mainly regular but for which there is evidence that a (small) chaotic part is present).
O. Bohigas, P. Leboeuf, M.-J. Sanchez, 'On the distribution of the total energy of a system of non-interacting fermions: random matrix and semiclassical estimates', Physica D 131 (1999) 186-204
O. Bohigas, P. Leboeuf, M.-J. Sanchez, 'Spectral spacing correlations for chaotic and disordered systems', Foundations of Phys., 31 (2001) 489-517
P. Leboeuf, A.G. Monastra, O. Bohigas, 'The Riemannium', Regular and Chaot. Dyn. 6 (2001) 205-210
O. Bohigas, P.Leboeuf, 'Nuclear masses: evidence of chaos-order coexistence', Phys. Rev. Lett. 88 (2002) 092502-1-4