Spectral decomposition of some non-self-adjoint operators

Abstract

We consider non-self-adjoint operators in Hilbert spaces of the form H = H0 + CW C, where H0 is self-adjoint, W is bounded and C is a metric operator, C bounded and relatively compact with respect to H0. We suppose that C(H0 − 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 ∈ σess(H) such that C(H ± 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 set of vectors ϕ such that e ±itH ϕ → 0 as t → ∞, coincide with the 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 Hac(H) = Hp(H * ) ⊥ , where Hp(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 Schrödinger operators with complex potentials.