Spectral decomposition of some non-self-adjoint operators

Abstract

We consider non-self-adjoint operators in Hilbert spaces of the form H = H 0 + CW 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 ∈ C \ R. We define the spectral singularities of H as the points of the essential spectrum λ ∈ σ ess (H) such that C(H − λ ± iε) −1 CW does not have a limit as ε → 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 ±itH φ → 0 as t → ∞, coincide with the finite linear combinaisons of generalized eigenstates of H corresponding to eigenvalues λ ∈ C, ∓ 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.