Bessel-Type Operators with an Inner Singularity

Brown, B. M.; Langer, H.; Langer, Matthias

doi:10.1007/s00020-012-2023-3

Cited by 5 publications

(2 citation statements)

References 26 publications

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“…3. By a celebrated result by DeBrange every classical Nevanlinna function appears as TitchmarshWeyl coefficient of a classical (two-dimensional) canonical system, this is, a boundary value problems of the form [15,31] for a model in a Pontryagin space and [50] and references therein for a model in a Hilbert space. 6.…”

Section: Generalized Nevanlinna Familiesmentioning

confidence: 99%

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2014

Operator Theory

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This article gives an introduction and short overview on generalized Nevanlinna functions, with special focus on asymptotic behavior and its relation to the operator representation. Introduction: Classical Nevanlinna FunctionsGeneralized Nevanlinna functions (scalar, matrix-or operator-valued) are functions that are meromorphic in CnR satisfying certain symmetry and sign conditions. They appeared first in connection with self-adjoint operators and relations in Pontryagin spaces (in [46] for scalar and [47] for matrix-valued functions) and have become an important tool in extension theory, e.g., for differential operators in connection with eigenvalue dependent boundary conditions. This survey is started with a short review of old and well-known facts about classical Nevanlinna functions (also known as Herglotz-, Pick-, or R-functions), which build a basis for the indefinite generalizations.Recall that a function q that maps the upper half plane C C holomorphically into C C [R is called a Nevanlinna function, q 2 N 0 .C/. These functions, which appear in many applications, e.g., as Titchmarsh-Weyl coefficients in Sturm-Liouville problems, are very well studied objects. In particular, such a function admits an integral representation ([37,58,59], presented in the following form by Cauer [16]; see also [1]):where a 2 R, b 0, and a positive Borel measure with R R d .t/ 1Ct 2 < 1. More abstractly, for every such function there exists a Hilbert space K; OE ; , a self-adjoint linear relation (i.e., a multi-valued operator) A in K and an element v in K such that with some z 0 2 %.A/ the function q can be written as q.z/ D q.z 0 / C .z z 0 / I C .z z 0 /.A z/ 1 v; v K :E-mail: luger@math.su.se Page 1 of 24Operator Theory DOI 10.1007/978-3-0348-0692-3_35-1 © Springer Basel 2015 In particular, for b D 0 one can choose K D L 2 , where is the measure in the integral representation (1), then A is the operator of multiplication by the independent variable. If b ¤ 0 then A is not an operator, but a relation with one-dimensional multi-valued part (see, e.g., [32] for the theory of linear relations and for the details of this representation, e.g., [53]).The limit behavior of q at the real line can be deduced directly from the above representations. In particular, lim z O !˛.˛ z/q.z/, where lim z O !˛d enotes the non-tangential limit to˛2 R, always exists and is zero or positive. Here the second case is equivalent to˛being an eigenvalue of the minimal representing relation A in (2), or, equivalently, f˛g ¤ 0.In several examples, however, e.g., in connection with singular Sturm-Liouville problems (see, e.g., [31,50,52]), there appear functions that are not as described above, but belong to a so-called generalized Nevanlinna class. Such functions may have non-real poles and the limit lim z O !˛.˛ z/q.z/ does not necessarily exist. However, this exceptional behavior can appear at finitely many points only, which relies on the fact that these functions admit operator representations of the form (2) but in Pontryagin spaces....

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Section: Generalized Nevanlinna Familiesmentioning

confidence: 99%

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2014

Operator Theory

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“…The determination of the Kreȋn-von Neumann extension and other nonnegative extensions can be found in [212,350]. For singular perturbations associated with Sturm-Liouville operators, see for instance [331] and the later papers [9,28,29,124,171,182,282,315,316,479,480,481,482,507,531,532,549,550]; for δ-point interactions we refer to [8,293]. Special properties of the Titchmarsh-Weyl coefficient have been studied in many papers; we just mention [130,292,366,367].…”

Section: Notes On Chaptermentioning

confidence: 99%

Boundary Value Problems, Weyl Functions, and Differential Operators

Behrndt

Hassi

Snoo

2020

Monographs in Mathematics

140

227

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The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.

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Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Luger

2015

Operator Theory

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Bessel-Type Operators with an Inner Singularity

Cited by 5 publications

References 26 publications

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

Boundary Value Problems, Weyl Functions, and Differential Operators

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

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