Spectral decomposition of some non-self-adjoint operators

Faupin, Jérémy; Frantz, Nicolas

Annales Henri Lebesgue, Tome 6 (2023), pp. 1115-1167

We consider non-self-adjoint operators in Hilbert spaces of the form $H=H_0+CWC$, where $H_0$ is self-adjoint, $W$ is bounded and $C$ is bounded and relatively compact with respect to $H_0$. We suppose that $C$ is a metric operator and that $C{(H_0-z)}^{-1}C$ is uniformly bounded in $z\in ℂ\setminus ℝ$. We define the spectral singularities of $H$ as the points of the essential spectrum $\lambda \in {\sigma }_{\mathrm{ess}}\left(H\right)$ such that $C{(H-\lambda \pm i\epsilon )}^{-1}CW$ does not have a limit as $\epsilon \to 0^{+}$. We prove that the spectral singularities of $H$ are in one-to-one correspondence with the eigenvalues, associated to resonant states, of an extension of $H$ to a larger Hilbert space. Next, we show that the asymptotically disappearing states for $H$, i.e. the vectors $\phi$ such that $e^{\pm itH}\phi \to 0$ as $t\to \infty$, coincide with the finite linear combinaisons of generalized eigenstates of $H$ corresponding to eigenvalues $\lambda \in ℂ$, $\mp Im\left(\lambda \right)>0$. Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies ${\mathscr{H}}_{\mathrm{ac}}\left(H\right)={\mathscr{H}}_{\mathrm{p}}{\left(H^{*}\right)}^{\perp }$, where ${\mathscr{H}}_{\mathrm{p}}\left(H^{*}\right)$ stands for the point spectral subspace of $H^{*}$. We thus obtain a direct sum decomposition of the Hilbert spaces in terms of spectral subspaces of $H$. One of the main ingredients of our proofs is a spectral resolution formula for a bounded operator $r\left(H\right)$ regularizing the identity at spectral singularities. Our results apply to Schrödinger operators with complex potentials.

Nous considérons une classe d’opérateurs non auto-adjoints sur un espace de Hilbert, de la forme $H=H_0+CWC$, où $H_0$ est auto-adjoint, $W$ est borné et $C$ est borné et relativement compact par rapport à $H_0$. On suppose que $C$ est un opérateur métrique et que $C{(H_0-z)}^{-1}C$ est uniformément borné pour $z\in ℂ\setminus ℝ$. Nous définissons les singularités spectrales de $H$ comme les points du spectre essentiel $\lambda \in {\sigma }_{\mathrm{ess}}\left(H\right)$ tels que $C{(H-\lambda \pm i\epsilon )}^{-1}CW$ n’a pas de limite quand $\epsilon \to 0^{+}$. Nous prouvons que les singularités spectrales de $H$ sont en bijection avec les valeurs propres associées à des états résonants d’une extension de $H$ à un espace de Hilbert plus gros. Ensuite, nous montrons que les états qui disparaissent à l’infini pour $H$, c’est à dire les $\phi$ tels que $e^{\pm itH}\phi \to 0$ quand $t\to \infty$, coïncident avec les vecteurs propres généralisés de $H$ associés à des valeurs propres $\lambda \in ℂ$, $\mp Im\left(\lambda \right)>0$. Finalement, nous définissons le sous-espace spectral absolument continu de $H$ et montrons qu’il satisfait ${\mathscr{H}}_{\mathrm{ac}}\left(H\right)={\mathscr{H}}_{\mathrm{p}}{\left(H^{*}\right)}^{\perp }$, où ${\mathscr{H}}_{\mathrm{p}}\left(H^{*}\right)$ est le sous-espace spectral ponctuel de l’opérateur adjoint $H^{*}$. Nous obtenons ainsi une décomposition en somme directe de l’espace de Hilbert en terme de sous-espaces spectraux de $H$. L’un des arguments principaux de nos preuves est une formule de résolution spectrale pour un opérateur borné $r\left(H\right)$ régularisant l’opérateur identité au voisinage des singularités spectrales. Nos résultats s’appliquent à des opérateurs de Schrödinger avec des potentiels complexes.

Classification : 47A10, 47A60, 47A65, 47B28, 81Q12, 81U24
Keywords: Non-self-adjoint operators, Spectral theory, Spectral singularities, Resonances, Schrödinger operators

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