Semiclassical sum rules for matrix elements and response functions in chaotic and in integrable quantum billiards

It is shown that expectation values and transition matrix elements in class ically chaotic quantum systems may not fluctuate randomly, since features o f the short-time classical dynamics significantly affect the fluctuations. We analyze semiclassical sum rules constraining expectation values and tran sition matrix elements in classically chaotic and integrable quantum system s. We show that these sum rules exhibit a wealth of interesting structures (resonances and oscillatory contributions) as well as interesting propertie s, such as their asymptotic decay. It is shown how these properties can be explained semiclassically, in terms of periodic and quasiperiodic classical motion. In particular, we analyze how phase-space inhomogeneities in chaot ic systems give rise to localization of wave functions and hence to excepti onally large matrix elements. These are related to resonances in classical autocorrelation functions. As an example, we consider a family of billiards in two dimensions the classical dynamics of which ranges from integrable t o chaotic. [S1063-651X(99)06201-7].