Spectral theory of pseudo-ergodic operators

We define a class of pseudo-ergodic non-self-adjoint Schrodinger operators acting in spaces l(2)(X) and prove some general theorems about their spectr al properties.; We then apply these to study the spectrum of a non-self-adj oint Anderson model acting on l(2)(Z), and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of lo calized spectrum for such operators.