Spectral and Transport Theory of Schrodinger Operators

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This research involves localization type effects for ergodic Schrodingeroperators, the general study of spectra and wave functions ofmultidimensional Schrodinger operators, and also of the transportphenomena of quasiperiodic operators and two-dimensional magneticoperators in the integer quantum Hall regime. An important objective is todevelop nonperturbative methods of proving localization type effects forSchrodinger operators with deterministic potentials. The other goal is tostudy the relation between spectral and quantum-dynamical properties andbehavior of the generalized eigenfunctions, particularly outside thelocalization range and in the multidimensional case. Another goal of theproposed research is to study singular continuous spectrum that exhibitscritical behavior and/or anomalous transport, particularly for models whereit appears for critical values (or intervals) of the parameters. The proposed research is centered around the fundamental propertiesof disordered systems that serve as models of systems with impurities.Deterministic, particularly quasiperiodic, potentials are most often usedto model quasicrystals. In order to be able to understand much of theexperimental data on quasicrystals, it is particularly important toinvestigate the transport coefficients like the electrical and heatconductivities of the microscopic models. Such an understanding is mosthelpful for finding new materials with desired physical properties. Thismay lead to various industrial applications (the first one nowadays beingthe covering of pans replacing the conventional Tefal film). Disorderedsystems are also used in modeling many other micro and macro effects: fromquantum localization to earthquakes. Our research concerns the anomalousspectral and diffusive properties of quasiperiodic and other deterministicstructures. The quantum Hall effect is since 1985 used by the NationalBureau of Standards to define the Fine Structure Constant (and hence theelectrical charge of an electron). It is still not well understood why theexperiment can be reproduced with a relative error of only $10^{-8}$. Ourresearch is concerned with a microscopic theory of the quantum Hall effectthat is aimed at getting deeper insights of this phenomena.