Scattering theory for some non-self-adjoint operators

Abstract

We consider a non-self-adjoint $H$ acting on a complex Hilbert space. We suppose that $H$ is of the form $H=H_0+CWC$ where $C$ is a bounded, positive definite and relatively compact with respect to $H_0$, and $W$ is bounded. We suppose that $C{(H_0-z)}^{-1}C$ is uniformly bounded in $z\in ℂ\setminus ℝ$. We define the regularized wave operators associated to $H$ and $H_0$ by $W_{\pm }(H,H_0):={\mathrm{\text{s-lim}}}_{t\to \infty }{e}^{\pm itH}{r}_{\mp }(H){\mathrm{\Pi }}_{\mathrm{\text{p}}}{(H^{\star })}^{\perp }{e}^{\mp it{H}_0}$ where ${\mathrm{\Pi }}_{\mathrm{\text{p}}}(H^{\star })$ is the projection onto the direct sum of all the generalized eigenspaces associated to eigenvalues of $H^{\star }$ and $r_{\mp }$ is a rational function that regularizes the “incoming/outgoing spectral singularities” of $H$. We prove the existence and study the properties of the regularized wave operators. In particular, we show that they are asymptotically complete if $H$ does not have any spectral singularity.