Scattering theory for some non-self-adjoint operators

TL;DR

This paper develops a scattering theory framework for certain non-self-adjoint operators, defining regularized wave operators and proving their existence and asymptotic completeness under specific spectral conditions.

Contribution

It introduces a novel approach to scattering theory for non-self-adjoint operators using regularized wave operators and analyzes their properties.

Findings

Existence of regularized wave operators for the class of operators considered.

Asymptotic completeness established when spectral singularities are absent.

Provides conditions ensuring the regularized wave operators are well-defined and complete.

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 $W_\pm(H,H_0):=\displaystyle\mathbb{s}-\lim_{t\rightarrow\infty} e^{\pm itH}r_\mp(H)\Pi_\mathrm{p}(H^\star)^\perp e^{\mp itH_0}$ 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.