Boundary conditions at infinity for difference equations of limit-circle type

Welstead, Stephen T.

doi:10.1016/0022-247x(82)90112-3

Cited by 10 publications

(4 citation statements)

References 6 publications

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“…See for instance [9] (addenda and problems to Chapter 1), [19], [27], [110] (Appendix E), [132], [135], 217 [188]. The characterization of boundary conditions in the limit circle case can be found in [242], [243] (however, a much more cumbersome approach is used there). A paper which focuses on the contributions by Krein is [186].…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…We assume that the minimal symmetric operator L min has deficiency index (1, 1), so that the Weyl-Hamburger limit-circle case holds for the matrix J (see [1][2][3][4][5][6][7][8][9][10][11][12][13]). Since L min has deficiency index (1, 1), P (λ) and Q (λ) belong to 2 In the following, we shall assume that L min is positive definite with lower bound μ > 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…[1][2][3][4] At the same time, these matrices can be perceived as second-order difference (or discrete Sturm-Liouville) operators. Also investigations including the studies of the deficiency indices of the symmetric operators generated by infinite Jacobi matrices, [1][2][3][4][5][6][7][8] self-adjoint and non-self-adjoint (dissipative and accumulative) extensions of symmetric operators have been carried in [3,6,[9][10][11][12][13]. Besides, Friedrichs and Krein-von Neumann extensions of positive definite symmetric operators generated by these matrices have been the priorities of [11,12].…”

Section: Introductionmentioning

confidence: 99%

Extensions of symmetric infinite Jacobi operator

Allahverdiev

2013

Linear and Multilinear Algebra

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A space of positive boundary values is constructed for a positive definite minimal symmetric operator generated by an infinite Jacobi matrix in limit-circle case. A description of all solvable, maximal dissipative (accumulative) solvable, self-adjoint solvable, positive definite self-adjoint and non-positive definite self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity.Keywords: infinite Jacobi matrix; symmetric operator; space of positive boundary values; solvable extensions; positive definite self-adjoint and non-self-adjoint extensions; maximal dissipative operator

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“…We assume that the minimal symmetric operator L 0 has defect index (1, 1), so that the Weyl limit-circle case holds for the expression y (see [1]- [4], [6], [16]). Since L 0 has defect index (1, 1), P (λ) and Q(λ) belong to 2 w ( ) for all λ ∈ .…”

Section: Extensions Of the Symmetric Operator Generated By An Infinitmentioning

confidence: 99%

Extensions, Dilations and Functional Models of Infinite Jacobi Matrix

Allahverdiev

2005

Czech Math J

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Boundary conditions at infinity for difference equations of limit-circle type

Cited by 10 publications

References 6 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

Extensions of symmetric infinite Jacobi operator

Extensions, Dilations and Functional Models of Infinite Jacobi Matrix

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