DILATIONS, MODELS, SCATTERING AND SPECTRAL PROBLEMS OF 1D DISCRETE HAMILTONIAN SYSTEMS

In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at +/-infinity) and acting in the Hilbert space l(Omega)(2)(Z; C-2) (Z := {0, +/- 1, +/- 2, ...}) are considered. We consider two classes dissipative operators with separated boundary conditions both at -infinity and infinity. For each of these cases we establish a self-adjoint dilation of the dissipative operator and construct the incoming and outgoing spectral representations that makes it possible to determine the scattering function (matrix) of the dilation. Further a functional model of the dissipative operator and its characteristic function in terms of the Weyl function of a selfadjoint operator are constructed. Finally we show that the system of root vectors of the dissipative operators are complete in the Hilbert space l(Omega)(2)(Z; C-2).