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Explicit formulas for the analytic extensions of the scattering matrix andthe time delay of a quasi-one-dimensional discrete Schr\"odinger operator witha potential of finite support are derived. This includes a careful analysis ofthe band edge singularities and allows to prove a Levinson-type theorem. Themain algebraic tool are the plane wave transfer matrices.

Analyticity properties of the scattering matrix for matrix Schr\"odinger operators on the discrete line