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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 473461, 16 pages
http://dx.doi.org/10.1155/2012/473461
Research Article

Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases

Department of Mathematics, Suleyman Demirel University, 32260 Isparta, Turkey

Received 4 January 2012; Accepted 11 April 2012

Academic Editor: Yuming Shi

Copyright © 2012 Bilender P. Allahverdiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space 2 𝑤 ( ) ( : = { 0 , ± 1 , ± 2 , } ), that is, the extensions of a minimal symmetric operator with defect index ( 2 , 2 ) (in the Weyl-Hamburger limit-circle cases at ± ). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at ” and “dissipative at .” In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators.