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 ±∞) and acting in the hilbert space ℓ_{ω}²(z;c²) (z:={0,±1,±2,...}) are considered. we consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. for each of these cases we establish a selfadjoint 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_{ω}²(z;c²).