We present a method to calculate the excitonic spectra of all direct semiconductors with a complex valence band structure. The Schrödinger equation is solved using a complete basis set with Coulomb Sturmian functions. This method also allows for the computation of oscillator strengths. Here we apply this method to investigate the impact of the valence band structure of cuprous oxide (Cu2O) on the yellow exciton spectrum. Results differ from those of J. Thewes et al. [Phys. Rev. Lett. 115, 027402 (2015)]; the differences are discussed and explained. The difference between the second and third Luttinger parameter can be determined by comparisons with experiments, however, the evaluation of all three Luttinger parameters is not uniquely possible. Our results are consistent with band structure calculations. Considering also a finite momentum K of the center of mass, we show that the large K-dependent line splitting observed for the 1S exciton state by G. Dasbach et al. [Phys. Rev. Lett. 91, 107401 (2003)] is not related to an exchange interaction but rather to the complex valence band structure of Cu2O.

According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wavefunctions of classically ergodic systems tend to the microcanonical density on the energy shell. We here develop a semiclassical theory that relates the rate of approach to the decay of certain classical uctuations. For uniformly hyperbolic systems we nd that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length TH , and inversely proportional to T 2 H , where TH = h is the Heisenberg time, being the mean density of states. Since for these systems the classical variance increases linearly with TH , the variance of the matrix elements decays like 1=TH. For non-hyperbolic systems, like Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic eld.