Spectral decomposition of some non-self-adjoint operators

Abstract

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 a metric operator, $C$ bounded and relatively compact with respect to $H_0$. We suppose that $C(H_0-z)^{-1}C$ is uniformly bounded in $z\in\mathbb{C}\setminus\mathbb{R}$. We define the spectral singularities of $H$ as the points of the essential spectrum $\lambda\in\sigma_{\mathrm{ess}}(H)$ such that $C(H\pm i\varepsilon)^{-1}CW$ does not have a limit as $\varepsilon\to0^+$. 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 set of vectors $\varphi$ such that $e^{\pm itH}\varphi\to0$ as $t\to\infty$, coincide with the generalized eigenstates of $H$ corresponding to eigenvalues $\lambda\in\mathbb{C}$, $\mp\mathrm{Im}(\lambda)>0$. Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies $\mathcal{H}_{\mathrm{ac}}(H)=\mathcal{H}_{\mathrm{p}}(H^*)^\perp$, where $\mathcal{H}_{\mathrm{p}}(H^*)$ stands for the point spectrum 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 Schr\"odinger operators with complex potentials.