Scattering theory for some non-self-adjoint operators

Abstract

We consider a non-self-adjoint [Formula: see text] acting on a complex Hilbert space. We suppose that [Formula: see text] is of the form [Formula: see text] where [Formula: see text] is a bounded, positive definite and relatively compact with respect to [Formula: see text], and [Formula: see text] is bounded. We suppose that [Formula: see text] is uniformly bounded in [Formula: see text]. We define the regularized wave operators associated to [Formula: see text] and [Formula: see text] by [Formula: see text] where [Formula: see text] is the projection onto the direct sum of all the generalized eigenspaces associated to eigenvalues of [Formula: see text] and [Formula: see text] is a rational function that regularizes the “incoming/outgoing spectral singularities” of [Formula: see text]. We prove the existence and study the properties of the regularized wave operators. In particular, we show that they are asymptotically complete if [Formula: see text] does not have any spectral singularity.