Approach to ergodicity in quantum wave functions

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 fluctuations. For uniformly hyperbolic systems we find 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 $T_H$, and inversely proportional to $T_H^2$, where $T_H=h\bar\rho$ is the Heisenberg time, $\bar\rho$ being the mean density of states. Since for these systems the classical variance increases linearly with $T_H$, the variance of the matrix elements decays like $1/T_H$. 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 field.Comment: 11 pages postscript and 4 figures in two files, tar-compressed and uuencoded using uufiles, to appear in Phys Rev E. For related papers, see http://www.icbm.uni-oldenburg.de/icbm/kosy/ag.htm

Minisymposium: "Degeneracies and Singularities in PDEs" nell'ambito del congresso: "8th European Conference on Elliptic and Parabolic Problems"

It is our intention to present some recent results on linear PDEs involving singular or degenerate operators acting in smooth domains as well as operators with smooth coefficients acting in irregular domains. In particular we address elliptic and parabolic boundary value problems in which fractal and euclidean interactions are relevant, Fractional Laplace operators and sharp Hardy- Sobolev inequalities. Stochastical aspects should be included

Contribution à l'étude de la genèse des minéraux argileux dans les bassins sédimentaires triasiques

In the present paper, an analysis of the clay minerals different genesis possibilities, in the spanish sedimentary Triassic basins, is made. The influence of the geomorphological evolution of continental reliefs on the inherited minerals' nature, is shown, as well as diagenetic action on sediments, producing aggradation phenomena, inverse to the one produced by chemical evolution of sedimentary basins.Dans ce travail, on fait une analyse des différents types de genèse des minéraux argileux dans des bassins sédimentaires triasiques espagnols. On montre l'influence que l'évolution géomorphologique des reliefs continentaux peut avoir sur la nature des minéraux hérités. On explique aussi comment la diagenèse peut agir sur les sédiments en provoquant des phénomènes dégradation inverses des mécanismes provoqués par l'évolution chimique des bassins sédimentaires.Caballero M.A., Martin vivaldi J.L. Contribution à l'étude de la genèse des minéraux argileux dans les bassins sédimentaires triasiques. In: Bulletin du Groupe français des argiles. Tome 26, fascicule 2, 1974. pp. 229-237

A pointwise regularity theory for the two-obstacle problem

TIB Hannover: RO 5389(24) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman