Reflection of microwaves from a cavity is measured in a frequency domain where the underlying classical chaotic scattering leaves a clear mark on the wave dynamics. We check the hypothesis that the fluctuations of the S matrix can be described in terms of parameters characterizing the chaotic classical scattering. Absorption of energy in the cavity walls is shown to significantly affect the results, and is linked to time-domain properties of the scattering in a general way. We also show that features whose origin is entirely due to wave dynamics (e.g., the enhancement of the Wigner time delay due to timereversal symmetry) coexist with other features which characterize the underlying classical dynamics.

We discuss two-point correlations of the actions of classical periodic orbits in chaotic systems. For systems where the semiclassical trace formula is exact and the spectral statistics follow random matrix theory, there exist nontrivial correlations between actions, which we express in a universal form. We illustrate this result with the analogous problem of the pair correlations between prime numbers. We also report on numerical studies of three chaotic systems where the semiclassical trace formula is only approximate, but nevertheless these unexpected action correlations are observed.PACS numbers: 05.45.+b, 03.65.Sq Semiclassical trace formulas relate the quantum spectral density to classical periodic orbits [1]. In this Letter we use this link to express a classical two-point correlation function, involving the actions and stabilities of the periodic orbits of chaotic systems, in terms of a two-point statistic of the quantum energy spectrum. We first consider systems for which the trace formulas are exact. Assuming that the spectral fluctuations follow the predictions of random matrix theory (RMT) [2,3], we derive a universal expression for the classical correlation function. This expression represents in some cases a tendency towards action repulsion. Studying in exactly the same way the correlations between pairs of prime numbers, i.e., assuming that the zeros of the Riemann zeta function follow RMT, we get a correlation which is consistent with the Hardy-Littlewood conjecture of number theory [4,5]. Surprisingly, we find that such correlations also occur in other chaotic systems which we studied numerically, and for which the semiclassical approximation (SCA) is not expected to be valid in the limit of long times. For definiteness we consider Hamiltonian flows with 2 freedoms and maps with 1 degree of freedom.Trace formulas express the spectral density by the Selberg-Gutzwiller sum [1], d{E) ^diE)-^dosM) -d{E)-\--^^SpTp h^r |det(M;-/)|'/2 exp / ^ 171 ^rS,--rv,where d{E) is the mean level density, and the summation is over primitive periodic orbits p and repetitions r (both positive and negative). Sp, Tp, gp, Mp, and Vp are the action, period, multiplicity, monodromy matrix, and Maslov index of the pth orbit, respectively. We label the preexponential factor in (1) by Aj, where j signifies the pair ip,r), and define Sj=rSp, Vj~rvp, and Tj=rTp,

We derive a semiclassical secular equation which applies to quantivd (compact) billiards of any shape. Our approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists bath on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization d e . We obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. We compare our result lo secular equations which were recently derived by other means, and provide some numerical data which illustrate our method when applied to the quantization of the Sinai billiard.