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• 1. Behrndt, Jussi
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On the negative squares of a class of self-adjoint extensions in Krein spaces2013Inngår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 286, nr 2-3, s. 118-148Artikkel i tidsskrift (Fagfellevurdert)

A description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function tau (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued D-K(nxn) -functions is introduced and, in particular, the properties of the inverse of a certain D-K(2x2) -function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non-positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.

• 2. Borogovac, Muhamed
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Analytic characterizations of Jordan chains by pole cancellation functions of higher order2014Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, nr 11, s. 4499-4518Artikkel i tidsskrift (Fagfellevurdert)

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length l, we show that there exists a pole cancellation function of order at least l, and, moreover, the construction shows that it is of surprisingly simple form.

• 3. Ivanenko, Y.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Quasi-Herglotz functions and convex optimization2020Inngår i: Royal Society Open Science, E-ISSN 2054-5703, Vol. 7, nr 1, artikkel-id 191541Artikkel i tidsskrift (Fagfellevurdert)

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

• 4. Ivanenko, Yevhen
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Passive Approximation and Optimization Using B-splines2019Inngår i: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 79, nr 1, s. 436-458Artikkel i tidsskrift (Fagfellevurdert)

A passive approximation problem is formulated where the target function is an arbitrary complex-valued continuous function defined on an approximation domain consisting of a finite union of closed and bounded intervals on the real axis. The norm used is a weighted L-p-norm where 1 <= p <= infinity. The approximating functions are Herglotz functions generated by a measure with Holder continuous density in an arbitrary neighborhood of the approximation domain. Hence, the imaginary and the real parts of the approximating functions are Holder continuous functions given by the density of the measure and its Hilbert transform, respectively. In practice, it is useful to employ finite B-spline expansions to represent the generating measure. The corresponding approximation problem can then be posed as a finite-dimensional convex optimization problem which is amenable for numerical solution. A constructive proof is given here showing that the convex cone of approximating functions generated by finite uniform B-spline expansions of fixed arbitrary order (linear, quadratic, cubic, etc.) is dense in the convex cone of Herglotz functions which are locally Holder continuous in a neighborhood of the approximation domain, as mentioned above. As an illustration, typical physical application examples are included regarding the passive approximation and optimization of a linear system having metamaterial characteristics, as well as passive realization of optimal absorption of a dielectric small sphere over a finite bandwidth.

• 5. Ivanenko, Yevhen
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Quasi-Herglotz functions and convex optimization2018Rapport (Annet vitenskapelig)

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modeling of non-passive systems.The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions, and we show that several of the important properties and modeling perspectives of Herglotz functions are inherited by the new set of quasi-Herglotz functions.In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory.Numerical examples are included to demonstrate the modeling of a non-passive gain media formulated as a convex optimization problem,where the generating measure is modeled by using a finite expansion of B-splines and point masses.

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• 6.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
An Operator Theoretic Interpretation of the Generalized Titchmarsh-Weyl Coefficient for a Singular Sturm-Liouville Problem2011Inngår i: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 14, nr 2, s. 115-151Artikkel i tidsskrift (Fagfellevurdert)

In this article an operator theoretic interpretation of the generalized Titchmarsh-Weyl coefficient for the Hydrogen atom differential expression is given. As a consequence we obtain a new expansion theorem in terms of singular generalized eigenfunctions.

• 7.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On supersingular perturbations of non-semibounded self-adjoint operators2019Inngår i: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 81, nr 1, s. 195-223Artikkel i tidsskrift (Fagfellevurdert)

In this paper self-adjoint realizations of the formal expression A(alpha ):= A + alpha <phi, .> phi are described, where alpha is an element of R boolean OR {infinity}, the operator A is self-adjoint in a Hilbert space H and phi is a supersingular element from the scale space H--(n) (-2) (A) \H--(n) (-1) (A) for n >= 1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to describe the family (A(alpha)). It can be interpreted in a Hilbert space with a twisted version of Krein's formula, or with a more classical version of Krein's formula but in a Pontryagin space. Finally, we compare the two approaches in terms of the respective Q-functions.

• 8.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On supersingular perturbations of not necessarily semibounded self-adjoint operatorsManuskript (preprint) (Annet vitenskapelig)
• 9.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior2015Inngår i: Operator Theory / [ed] Daniel Alpay, Springer, 2015, s. 345-371Kapittel i bok, del av antologi (Fagfellevurdert)

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.

• 10.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
A characterization of Herglotz–Nevanlinna functions in two variables via integral representations2017Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 55, nr 1, s. 199-216Artikkel i tidsskrift (Fagfellevurdert)

We derive an integral representation for Herglotz-Nevanlinna functions in two variables, which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. Further properties of the class of representing measures are also discussed.

• 11.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
An integral representation for Herglotz-Nevanlinna functions in several variables2017Rapport (Annet vitenskapelig)

In this article, a characterization of the class of Herglotz-Nevanlinna functions in n variables is given in terms of an integral representation. Furthermore, the properties of the class of representing measures are discussed in detail. Symmetry properties induced by the integral representation are also investigated.

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• 12.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Geometric properties of measures related to holomorphic functions having positive imaginary or real partManuskript (preprint) (Annet vitenskapelig)

In this paper, we study the properties of a certain class of Borel measures on R^n that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures can not lie within particular geometric regions, \eg strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to  functions mapping the unit polydisk analytically into the right half-plane.

• 13.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Geometric Properties of Measures Related to Holomorphic Functions Having Positive Imaginary or Real Part2020Inngår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002XArtikkel i tidsskrift (Fagfellevurdert)

In this paper, we study the properties of a certain class of Borel measures on Rnthat arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures cannot lie within particular geometric regions, e.g., strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.

• 14.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Herglotz-Nevanlinna functions in several variablesManuskript (preprint) (Annet vitenskapelig)

In this article, a characterization of the class of Herglotz-Nevan\-linna functions in several variables is given in terms of an integral representation. The conditions on the representing measure are discussed in detail, and, furthermore, the properties of the symmetric extension of a Herglotz-Nevanlinna function are also investigated.

• 15.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Herglotz-Nevanlinna functions in several variables2019Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 472, nr 1, s. 1189-1219Artikkel i tidsskrift (Fagfellevurdert)

In this article, a characterization of the class of Herglotz-Nevanlinna functions in several variables is given in terms of an integral representation. The conditions on the representing measure are discussed in detail, and, furthermore, the properties of the symmetric extension of a Herglotz-Nevanlinna function are also investigated.

• 16.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
An Operator Theoretic Interpretation of the Generalized Titchmarsh–Weyl Function for Perturbed Spherical Schrödinger Operators2015Inngår i: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 9, nr 6, s. 1391-1410Artikkel i tidsskrift (Fagfellevurdert)

In this paper the operator theoretic interpretation of the generalized Titchmarsh–Weyl function is extended to the perturbed spherical Schrödinger operator.

• 17.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On the Weyl solution of the 1-dim Schrödinger operator with inverse fourth power potential2016Inngår i: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 180, nr 2, s. 295-303Artikkel i tidsskrift (Fagfellevurdert)

We consider the one dimensional Schrödinger operator with potential 1/x4" role="presentation">1/x4 on the half line. It is known that a generalized Titchmarsh–Weyl function can be associated to it. For other strongly singular potentials in some previous works it was possible to give an operator theoretic interpretation of this fact. However, for the present potential we show that such an interpretation does not exist.

• 18.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Asymptotics of the Weyl function for Schrodinger operators with measure-valued potentials2016Inngår i: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 179, nr 4, s. 603-613Artikkel i tidsskrift (Fagfellevurdert)

We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrodinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.

• 19.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Herglotz functions and applications in electromagneticsManuskript (preprint) (Annet vitenskapelig)

Herglotz functions inevitably appear in pure mathematics, mathematical physics, and engineering with a wide range of applications. In particular, they are the pertinent functions to model passive systems, and thus appear in modeling of electromagnetic phenomena in circuits, antennas, materials, and scattering. In this chapter, we review the basic theory of Herglotz functions and its applications to determine sum rules and physical bounds for passive systems.

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