Generalized resolvents and the boundary value problems for Hermitian operators with gaps
TL;DR: In this paper, a Hermitian operator A with gaps (αj, βj) (1 ⩽ j⩽ m ⩾ ∞) is studied and the self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap are described in terms of boundary conditions.
About: This article is published in Journal of Functional Analysis.The article was published on 1991-01-01 and is currently open access. It has received 598 citations till now. The article focuses on the topics: Extensions of symmetric operators & Hermitian matrix.
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368 citations
TL;DR: In this paper, the authors provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrixvalued Schrodinger and Dirac type operators.
Abstract: We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations.
Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.
291 citations
Additional excerpts
...…and difference operators, interpolation problems, and factorizations of matrix and operator functions [5], [14]–[16], [20], [24], [25], [27], [30], [35]– [38], [40], [47], [49]–[51], [65], [66], [69]–[80], [85], [89], [90], [96]–[99], [106]–[110], [116], [120], [122], [145], [146], inverse…...
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TL;DR: In this article, a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples is given, and a description of the spectra of selfadjoint extension in terms of the corresponding Krein maps (Weyl functions) is given.
Abstract: We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces and singular perturbations.
251 citations
Cites background or methods from "Generalized resolvents and the boun..."
...In [ 49 , 71, 88] one defines boundary triple only for the case when A ∗ is a closed densely...
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...A systematic theory of self-adjoint extensions in terms of boundary conditions, including the spectral analysis, was developed by Derkach and Malamud, who found, in particular, a nice relationship between the parameters of s elf-adjoint extensions and the Krein resolvent formula, and performed the spectral analysis in terms of the Weyl functions; we refer to the paper [ 49 ] summarizing this machinery and containing an extensive ......
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Book•
26 Jan 2012
TL;DR: In this article, the convergence results for star graphs and associated Laplacians were obtained for different Hilbert spaces and boundary triples, and two operators in different Hilbert space operators were compared.
Abstract: 1 Introduction.- 2 Graphs and associated Laplacians.- 3 Scales of Hilbert space and boundary triples.- 4 Two operators in different Hilbert spaces.- 5 Manifolds, tubular neighbourhoods and their perturbations.- 6 Plumber's shop: Estimates for star graphs and related spaces.- 7 Global convergence results.
216 citations
TL;DR: In this paper, the concepts of boundary relations and the corresponding Weyl families are introduced, and fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : (H 2, J(H)) -> (H-2, J (H)), where Gamma need not be surjective and is even allowed to be multivalued.
Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
172 citations
Cites background from "Generalized resolvents and the boun..."
...1 of an ordinary boundary triplet for the case of a densely defined symmetric operator from [20] (see also [15], [28]) and the adaptation for the case of a nondensely defined symmetric operator leads to the following definition....
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...9) (see [15]), and this makes it difficult to investigate the appropriate spectral properties determined by the given boundary conditions....
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...In [14, 15] the concept of a Weyl function was associated to an ordinary boundary triplet as an abstract version of the so-called m-function appearing in boundary value problems for differential operators....
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...In this case the corresponding Weyl function is M(λ) = (A− √ A2 − λ)A; see [15]....
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...It was shown in [15], [28] that γ(·) and M(·) satisfy (4....
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References
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Book•
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836
19,846 citations
1,113 citations
Book•
01 Dec 1987
TL;DR: De Gruyter as discussed by the authors presents a detailed theory of inverse problems and methods of their solution for the Sturm-Liouville case, with a focus on quantum scattering theory.
Abstract: 01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. The interest in inverse problems of spectral analysis has increased considerably in recent years due to the applications to important non-linear equations in mathematical physics. This monograph is devoted to the detailed theory of inverse problems and methods of their solution for the Sturm-Liouville case. Chapters 1--6 contain proofs which are, in many cases, very different from those known earlier. Chapters 4--6 are devoted to inverse problems of quantum scattering theory with attention being focused on physical applications. Chapters 7--11 are based on the author's recent research on the theory of finite- and infinite-zone potentials. A chapter discussing the applications to the Korteweg--de Vries problem is also included. This monograph is important reading for all researchers in the field of mathematics and physics.
1,022 citations