Research PaperNo Access

Scattering theory for some non-self-adjoint operators

Laboratoire Analyse Géométrie Modélisation, CY Cergy Paris Université, 95302 Cergy Pontoise, France

https://doi.org/10.1142/S0129055X24500235Cited by:0 (Source: Crossref)

Abstract

We consider a non-self-adjoint $H$ acting on a complex Hilbert space. We suppose that $H$ is of the form $H = H_{0} + C W C$ 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 $z \in ℂ ∖ ℝ$ . We define the regularized wave operators associated to $H$ and $H_{0}$ by $W_{\pm} (H, H_{0}) : = {s-lim}_{t \to \infty} e^{\pm i t H} r_{\mp} (H) Π_{p} {(H^{⋆})}^{⊥} e^{\mp i t H_{0}}$ where $Π_{p} (H^{⋆})$ is the projection onto the direct sum of all the generalized eigenspaces associated to eigenvalues of $H^{⋆}$ 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.

Keywords:

AMSC: 35P25, 34L05, 34L40, 35B34

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