Pacific Institute for the Mathematical Sciences - PIMS

Localization of the eigenfunctions of the one-dimensional Schrödinger operator in the presence of random potentials

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.

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