Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases
We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ℓ2𝑤(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 ±∞). We investigate...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/473461 |
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| _version_ | 1832562817327169536 |
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| author | Bilender P. Allahverdiev |
| author_facet | Bilender P. Allahverdiev |
| author_sort | Bilender P. Allahverdiev |
| collection | DOAJ |
| description | We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ℓ2𝑤(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 ±∞). 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. |
| format | Article |
| id | doaj-art-71e4afde24d04456a58549dee5c82044 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-71e4afde24d04456a58549dee5c820442025-02-03T01:21:37ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/473461473461Nonself-Adjoint Second-Order Difference Operators in Limit-Circle CasesBilender P. Allahverdiev0Department of Mathematics, Suleyman Demirel University, 32260 Isparta, TurkeyWe consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space ℓ2𝑤(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 ±∞). 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.http://dx.doi.org/10.1155/2012/473461 |
| spellingShingle | Bilender P. Allahverdiev Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases Abstract and Applied Analysis |
| title | Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases |
| title_full | Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases |
| title_fullStr | Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases |
| title_full_unstemmed | Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases |
| title_short | Nonself-Adjoint Second-Order Difference Operators in Limit-Circle Cases |
| title_sort | nonself adjoint second order difference operators in limit circle cases |
| url | http://dx.doi.org/10.1155/2012/473461 |
| work_keys_str_mv | AT bilenderpallahverdiev nonselfadjointsecondorderdifferenceoperatorsinlimitcirclecases |