Damanik, David2011-07-252011-07-252010Kruger, Helge. "Positive Lyapunov exponent for ergodic Schrodinger operators." (2010) Diss., Rice University. <a href="https://hdl.handle.net/1911/62012">https://hdl.handle.net/1911/62012</a>.https://hdl.handle.net/1911/62012The discrete Schrodinger equation describes the behavior of a 1-dimensional quantum particle in a disordered medium. The Lyapunov exponent L( E) describes the exponential behavior of solutions at an energy E. Positivity of the Lyapunov exponent in a set of energies is an indication of absence of transport for the Schrodinger equation. In this thesis, I will discuss methods based on multiscale analysis to prove positive Lyapunov exponent for ergodic Schrodinger operators. As an application, I prove positive Lyapunov exponent for operators whose potential is given by evaluating an analytic sampling function along the orbit of a skew-shift on a high dimensional torus. The first method is based only on ergodicity, but needs to eliminate a small set of energies. The second method uses recurrence properties of the skew-shift, combined with analyticity to prove a result for all energies.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisTHESIS MATH. 2010 KRUGER