Georgia Institute of Technology College of Sciences

One dimensional discrete Schrödinger operators arise naturally in modeling the motion of quantum particles in a disordered medium. The medium is described by potentials which may naturally be generated by certain ergodic dynamics. We will begin with two classic models where the potentials are periodic sequences and i.i.d. random variables (Anderson Model). Then we will move on to quasi-periodic potentials, of which the randomness is between periodic and i.i.d models and the phenomena may become more subtle, e.g. a metal-insulator type of transition may occur. We will show how the dynamical object, the Lyapunov exponent, plays a key role in the spectral analysis of these types of operators.