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 ±∞1) and acting in the Hilbert space ℓ²Ω(ℤ,ℂ²) (ℤ:= {0,±1,±2, ...}) are considered. We deal with two classes of dissipative operators with separated boundary conditions both at -∞ and ∞. 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 ℓ²Ω(ℤ,ℂ²). [ABSTRACT FROM AUTHOR]