No abstract

We consider the optical absorption and emission spectra of excitons in two-dimensional semiconductors disordered through interface fluctuations. These spectra show a universal behavior exemplified by the fact that the offset of the spectral peaks (the Stokes shift) is proportional to their linewidths over a range of at least 2 orders of magnitude. We introduce a topographical theory of the exciton spectra which models such behavior in terms of statistical properties of a Gaussian random function. The coefficient of proportionality between the Stokes shift and the exciton absorption linewidth is found to be 7 = 2/\/67r In 2 = 0.553 by analysis and 0.6 by experiment.

The spectrum of Harper's equation (a model for Bloch electrons in a magnetic field) is a fractal Cantor set if the ratio P of the area of a unit cell to that of a fiux quantum is not a rational number. It has been conjectured that the second moment of an initially localized wave packet has a powerlaw growth of the form (z ) t ', where Dp is the box-counting dimension of the spectrum, and that Do --~. We present numerical results on the dimension of the spectrum and the spread of a wave packet indicating that these relationships are at best approximate. We also present heuristic arguments suggesting that there should be no general relationships between the dimension and the spread of a wave packet.

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We consider the dynamics of a complex quantum system subjected to a timedependent pefiurbotion, using a random matrix approach. The dynamics x e described by a diffusion constant chancteridng the spread of the probability distribution for the energy of a particle which was initially in an eigenstate. We discuss a System of stochastic differentid equations which are a model for the Schradinger equation written in an adiabatic basis. We examine the,dependence of the diffusion constant D on the rate of change of the pemrbation parameter, X. Our a!~alysis indiwtes that D c(X2, in a p e m e n t with the Kubo formula, up fo a critical velocity X'; for faster perturbations. the rate of diffusion is lower than that predicted from the Kubo formula. These predictions are confirmed in numerical experiments on a banded random matrix model. The implications of this result are discussed.