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- Title
Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases.
- Authors
Allahverdiev, Bilender P.
- Abstract
We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space lw² (Z) (Z:= {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 ±8). We investigate two classes of maximal dissipative operators with separated boundary conditions, called "dissipative at -8" and "dissipative at 8." 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.
- Subjects
NONSELFADJOINT operators; DIFFERENCE operators; LIMIT theorems; HILBERT space; S-matrix theory; EIGENVECTORS; STURM-Liouville equation
- Publication
Abstract & Applied Analysis, 2012, p1
- ISSN
1085-3375
- Publication type
Article
- DOI
10.1155/2012/473461