## 2009 Probability Seminar - 08

- Date: 04/01/2009

University of British Columbia

Localization of the eigenfunctions of the one-dimensional SchrÃ¶dinger operator in the presence of random potentials

We consider the one-dimensional discrete SchrÃ¶dinger 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.

3:00pm-4:00pm, WMAX 216