UC Irvine

We will study the spectrum of a discrete Schr ̈odinger operator called the periodic Anderson-Bernoulli operator. Because the operator is ergodic, we can use techniques in dynamical systems and apply Johnson’s Theorem to better understand this operator. These techniques involve study- ing the hyperbolic locus of SL(2, R) cocycles and the geometry of the hyperbolic locus in SL(2, R)n. This model has a spectrum that is completely pure-point, and there exist parameters such that the spectrum can be defined as the union of an infinite number of intervals, which is unexpected for multiple reasons. A result of Avila, Damanik, and Gorodetski [2] says that if the Anderson model has a background potential defined by a dynamical system with a continuous phase space, then such a result is impossible. In this model, the background potential is periodic, making that result not applicable. This thesis provides details of this work, as well as insight into where this work may progress.