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Information Journal Paper

Title

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

Pages

 Start Page 1553 | End Page 1571

Abstract

 In this paper, the maximal dissipative extensions of a symmetric singular 1D discrete Hamiltonian operator with maxi-mal deficiency indices (2, 2) (in limit-circle cases at ¥) and ¥ acting in the Hilbert space L2W(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 deter-mine 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 L2W(Z, C2).

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