The finite section method for dissipative operators

We show that for self-adjoint Jacobi matrices and Schrödinger operators, perturbed by dissipative potentials in ℓ 1 (N) and L 1 (0,∞) respectively, the finite section method does not omit any points of the spectrum. In the Schrödinger case two different approaches are presented. Many aspects of the proofs can be expected to carry over to higher dimensions, particularly for absolutely continuous spectrum