2009 Probability Seminar - 08

We consider the one-dimensional discrete Schrödinger operator (Hf)(x)=f(x-1)+f(x+1)+v(x)f(x), on the interval {1,2,...,N} with Dirichlet boundary conditions f(0)=f(N+1)=0. We assume that v(x) are independent random variables for a very small fraction of the sites x and nonrandom for the remaining sites. We discuss a mechanism responsible for the following localization phenomenon: for large N, outside a set of realizations of the potentials of very small probability, each eigenfunction of H decays exponentially. Our approach to localization is based on a recent method of Goldstein.