Scattering theory for some non-self-adjoint operators

Abstract

We consider a non-self-adjoint $H$ given as the perturbation of a self-adjoint operator $H_0$. 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 $z\in \mathbb{C}\setminus \mathbb{R}$. We define the regularized wave operators associated to $H$ and $H_0$ by $W_{\pm }(H,H_0):=s-\underset{t\to \infty }{\lim }{e}^{\pm itH}{r}_{\mp }(H){\Pi }_{\mathrm{p}}(H^{\star }{)}^{\perp }{e}^{\mp it{H}_0}$ where ${\Pi }_{\mathrm{p}}(H^{\star })$ is the projection onto the direct sum of all the generalized eigenspace associated to eigenvalue 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.

arXiv comment34 pages, 1 figure, the results proven rely on the material introduced in arxiv:2203.12406 which is recalled in this paper