Positive Lyapunov exponent for ergodic Schrodinger operators
Doctor of Philosophy
The 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.