Spectral estimates for the one-dimensional non-self-adjoint Anderson model

Abstract

We obtain bounds on the spectrum of the non-self-adjoint Anderson operator in one dimension by using higher order numerical ranges and are able to determine the spectrum completely in many cases. We also develop further some previously existing methods that allow us to prove that certain curves and regions are contained in the spectrum and provide numerical examples that suggest that these curves contained in the infinite volume spectrum have a strong bearing over the finite volume cases.