Dissipative discrete Hamiltonian systems

A space of boundary values is constructed for minimal symmetric operator, generated by discrete Hamiltonian system, acting in the Hilbert space l2A (ℤ E ⊕ E) (ℤ={0, ±1, ±2, …}, dim E = n < ∞) with deficiency indices (n, n) (in limit-circle case at ±∞ and limit point case at ∓∞). A description of all maximal dissipative, maximal accretive, and self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at ±∞. We investigate two classes of maximal dissipative operators with boundary conditions, called 'dissipative at −∞' and 'dissipative at ∞'. In each of these cases we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation in terms of the Titchmarsh-Weyl matrix-valued function of the self-adjoint operator. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of the scattering matrix of dilation. Finally, we prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.