Approximation of N κ ∞ -functions I: Models and Regularization

Dijksma, Aad; Luger, Annemarie; Shondin, Yuri

doi:10.1007/978-3-7643-8911-6_5

Cited by 10 publications

(6 citation statements)

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“…k, l ≤ 2k, ⌊ l 2 ⌋, l ≥ 2k + 1, l even, or, l odd and a l > 0, ⌊ l 2 ⌋ + 1, l ≥ 2k + 1, l odd and, a l < 0. For additional equivalent conditions we refer to Definition 2.5 in [12]. Given a generalized Nevanlinna function in N ∞ κ , the corresponding κ is given by the multiplicity of the generalized pole at ∞ which is determined by the facts that the following limits exist and take values as indicated: Again the limits can be replaced by nontangential ones.…”

Section: Appendix C Generalized Nevanlinna Functionsmentioning

confidence: 99%

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

Kostenko

Sakhnovich

2011

International Mathematics Research Notices

185

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We develop Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials such as perturbed spherical Schrödinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schrödinger operators.2000 Mathematics Subject Classification. Primary 34B20, 34L05; Secondary 34B24, 47A10.

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Section: Appendix C Generalized Nevanlinna Functionsmentioning

confidence: 99%

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

Kostenko

Sakhnovich

2011

International Mathematics Research Notices

185

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“…In connection with supersingular perturbations, the class N 1 Ä plays an important role; it consists of all functions q 2 N Ä for which 1 is the only generalized pole not of positive type; see, e.g., [26].…”

Section: Some Special Subclassesmentioning

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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“…<∞ previously appeared in many papers in the context of Sturm-Liouville equations with singular endpoints or singular perturbations, see [DLS], [DHS], [DKuS], [DLSZ], [DLuS1], [DLuS2], [DLuS3], [KuLu].…”

Section: The Class N (∞)mentioning

confidence: 99%