Spectral decomposition of some non-self-adjoint operators

Faupin, Jérémy; Frantz, Nicolas

doi:10.5802/ahl.185

Faupin, Jérémy; Frantz, Nicolas

Spectral decomposition of some non-self-adjoint operators

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

Metadata

DOI10.5802/ahl.185

Keywords Non-self-adjoint operators, Spectral theory, Spectral singularities, Resonances, Schrödinger operators

Abstract

We consider non-self-adjoint operators in Hilbert spaces of the form $H = H_{0} + C W C$ , 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 $z \in ℂ ∖ ℝ$ . We define the spectral singularities of $H$ as the points of the essential spectrum $λ \in σ_{ess} (H)$ such that $C {(H - λ \pm i ε)}^{- 1} C W$ does not have a limit as $ε \to 0^{+}$ . 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 $φ$ such that $e^{\pm i t H} φ \to 0$ as $t \to \infty$ , coincide with the finite linear combinaisons of generalized eigenstates of $H$ corresponding to eigenvalues $λ \in ℂ$ , $\mp Im (λ) > 0$ . Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies $ℋ_{ac} (H) = ℋ_{p} {(H^{*})}^{⊥}$ , where $ℋ_{p} (H^{*})$ 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 $r (H)$ regularizing the identity at spectral singularities. Our results apply to Schrödinger operators with complex potentials.

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