Spectral Theory of Pseudo-Ergodic Operators

We define a class of pseudo-ergodic non-self-adjoint Schr?dinger operators acting in spaces l ²(X) and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting onl ²(Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators. Received: 8 August 2000 / Accepted: 3 October 2000