Spectral decomposition of some non-self-adjoint operators
Annales Henri Lebesgue, Volume 6 (2023), pp. 1115-1167.


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


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. We define the spectral singularities of H as the points of the essential spectrum λσ ess (H) such that C(H-λ±iε) -1 CW does not have a limit as ε0 + . 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 φ0 as t, coincide with the finite linear combinaisons of generalized eigenstates of H corresponding to eigenvalues λ, Im(λ)>0. Finally, we define the absolutely continuous spectral subspace of H and show that it satisfies ac (H)= p (H * ) , where 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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