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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1085-3375 1687-0409 |