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 }$\pm \infty $ and acting in the Hilbert space $\ell _{\Omega }^{2}(\mathbb{Z}; ‎ ‎\mathbb{C}^{2})${ \ }$(\mathbb{Z}:=\{0,\pm 1,\pm 2,...\})${ \‎ ‎are considered.\ We deal with two classes of dissipative operators with‎ ‎separated boundary conditions both at }$-\infty ${ \ and }$\infty‎ .‎$‎ ‎ \ For each of these cases‎, ‎we establish a self-adjoint dilation\ of‎ ‎the dissipative operator and construct the incoming and outgoing spectral‎ ‎representations‎. ‎Then‎, ‎it becomes possible to determine the scattering function‎ ‎(matrix) of the dilation‎. ‎Further‎, ‎a functional model of the dissipative‎ ‎operator and its characteristic function in terms of the Weyl function of a‎ ‎self-adjoint operator are constructed‎. ‎Finally‎, ‎we show that the system of‎ ‎root vectors of the dissipative operators are complete in the Hilbert space‎ ‎$\ell _{\Omega }^{2}(\mathbb{Z};\mathbb{C}^{2})$‎.