Dilation, Model, Scattering and Spectral Problems of Second-Order Matrix Difference Operator

Abstract

In the Hilbert space ℓ 2 Ω (Z; E) (Z := {0,±1,±2, ...}, dim E = N < ∞), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at ±∞) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functional model of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved.