Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

We consider Schrödinger operators of the form H R = - d 2 / d x 2 + q + i γ χ [ 0 , R ] for large R > 0 , where q ∈ L 1 ( 0 , ∞ ) and γ > 0 . Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R .