## Abstract and Applied Analysis

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

Bilender P. Allahverdiev

#### Abstract

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ${\mathcal{l}}_{w}^{2}(\mathbb{Z})$ ($\mathbb{Z}$:$=\{0,{\pm}1,{\pm}2,\dots \}$), that is, the extensions of a minimal symmetric operator with defect index ($2,2$) (in the Weyl-Hamburger limit-circle cases at ${\pm}\infty$). We investigate two classes of maximal dissipative operators with separated boundary conditions, called “dissipative at $-\infty$” and “dissipative at $\infty$.” 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 473461, 16 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495742

Digital Object Identifier
doi:10.1155/2012/473461

Mathematical Reviews number (MathSciNet)
MR2935144

Zentralblatt MATH identifier
1242.47027

#### Citation

Allahverdiev, Bilender P. Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases. Abstr. Appl. Anal. 2012 (2012), Article ID 473461, 16 pages. doi:10.1155/2012/473461. https://projecteuclid.org/euclid.aaa/1355495742

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