Dynamics of Schroedinger Cocycles and Applications to Spectral Theory

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This project will investigate spectral problems with the help of dynamical systems tools. The object of study are Schroedinger operators whose potentials are obtained by sampling with a continuous function along the orbits of an ergodic transformation on a compact metric space. This framework covers many examples of interest, such as almost-periodic potentials and random potentials. The spectral properties of such operators are closely linked to the dynamical behavior of an energy-indexed family of SL(2,R)-valued cocycles over the given ergodic transformation. Of interest are in particular the Lyapunov exponents associated with these cocycles. The following spectral problems will be investigated: purely absolutely continuous spectrum for quasi-periodic potentials at small coupling for arbitrary irrational frequency, the genericity of Cantor spectrum for suitable classes of transformations and sampling functions, the irregularity of the Lyapunov exponent as a function of the energy, spectral phenomena for perturbed quasi-periodic potentials, and restrictions put on the potentials by the existence of absolutely continuous spectrum.Quantum mechanics is a fundamental branch of physics whose foundations were established during the first half of the twentieth century. The study of quantum mechanical phenomena in disordered environments has been an area of ongoing active study since the 1950's. A landmark paper was published by Anderson in 1958. He was awarded the Nobel Prize in Physics in 1977 for his work on the absence of diffusion for certain random lattice Hamiltonians. Another event of importance was the discovery of quasicrystals by Shechtman in 1982, which was reported in a 1984 paper he wrote jointly with Blech, Gratias and Cahn, and which caused a paradigm shift in crystallography and solid state physics. The mathematical study of electronic properties of disordered structures is carried out within the framework of ergodic Schroedinger operators. Since the potentials of these operators are defined dynamically, namely by sampling along the orbits of one or more ergodic transformations, it is quite natural that dynamical systems tools should prove to be useful in the study of such operators. The field has recently taken major leaps after a number of very talented young researchers from dynamical systems entered it. This has also lead to fruitful collaborations across the disciplines and there is promise for further success of these interactions.