| 1. |
- Behrndt, Jussi, et al.
(författare)
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On the negative squares of a class of self-adjoint extensions in Krein spaces
- 2013
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Ingår i: Mathematische Nachrichten. - : Wiley. - 0025-584X .- 1522-2616. ; 286:2-3, s. 118-148
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Behrndt, Jussi, et al.
(författare)
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On the number of negative eigenvalues of the Laplacian on a metric graph
- 2010
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Ingår i: Journal of Physics A: Mathematical and Theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 43:47
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Tidskriftsartikel (refereegranskat)abstract
- The number of negative eigenvalues of self-adjoint Laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure. This extends and complements earlier results by Kostrykin and Schrader (2006 Contemp. Math. 415 201-25).
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| 3. |
- Bernland, Anders, et al.
(författare)
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Sum rules and constraints on passive systems
- 2011
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Ingår i: Journal of Physics A: Mathematical and Theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 44:14
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Tidskriftsartikel (refereegranskat)abstract
- A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. A system in convolution form is fully described by its transfer function, and the class of Herglotz functions, holomorphic functions mapping the open upper half plane to the closed upper half plane, is closely related to the transfer functions of passive systems. Following a well-known representation theorem, Herglotz functions can be represented by means of positive measures on the real line. This fact is exploited in this paper in order to rigorously prove a set of integral identities for Herglotz functions that relate weighted integrals of the function to its asymptotic expansions at the origin and infinity. The integral identities are the core of a general approach introduced here to derive sum rules and physical limitations on various passive physical systems. Although similar approaches have previously been applied to a wide range of specific applications, this paper is the first to deliver a general procedure together with the necessary proofs. This procedure is described thoroughly, and exemplified with examples from electromagnetic theory.
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| 4. |
- Bernland, Anders, et al.
(författare)
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Sum rules and constraints on passive systems
- 2010
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. A system on convolution form is fully described by its transfer function, and the class of Herglotz functions, holomorphic functions mapping the open upper half plane to the closed upper half plane, is closely related to the transfer functions of passive systems. Following a well-known representation theorem, Herglotz functions can be represented by means of positive measures on the real line. This fact is exploited in this paper in order to rigorously prove a set of integral identities for Herglotz functions that relate weighted integrals of the function to its asymptotic expansions at the origin and infinity. The integral identities are the core of a general approach introduced here to derive sum rules and physical limitations on various passive physical systems. Although similar approaches have previously been applied to a wide range of specific applications, this paper is the first to deliver a general procedure together with the necessary proofs. This procedure is described thoroughly, and exemplified with examples from electromagnetic theory.
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| 5. |
- Bernland, Anders, et al.
(författare)
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Sum rules and constraints on passive systems with applications in electromagnetics
- 2010
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Ingår i: [Host publication title missing]. ; , s. 33-36
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Konferensbidrag (refereegranskat)abstract
- A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. In this paper there is a review of some results on linear, time translational invariant, continuous, causal and passive systems, where it turns out that Herglotz functions are related to the Fourier transform of the impulse response of such systems. Some well known facts of this function class is considered, and a set of integral identities and an outline of the proof of these are presented. The identities may be used to derive sum rules and constraints on various physical systems. The theory is illuminated with two examples from electromagnetics: the first revisits Fano’s matching equations, while the latter makes a link to the Kramers-Kronig relations and discusses physical limitations on metamaterials.
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| 6. |
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| 7. |
- Borogovac, Muhamed, et al.
(författare)
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Analytic characterizations of Jordan chains by pole cancellation functions of higher order
- 2014
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 267:11, s. 4499-4518
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Dijksma, Aad, et al.
(författare)
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Approximation of N∞ κ -functions I : Models and regularization
- 2008
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Ingår i: Spectral Theory in Inner Product Spaces and Applications - 6th Workshop on Operator Theory in Krein Spaces and Operator Polynomials, 2006. - 0255-0156 .- 2296-4878. - 9783764389109 ; 188, s. 87-112
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Konferensbidrag (refereegranskat)abstract
- The class N∞ κ consists of all generalized Nevanlinna functions N with κ negative squares for which the root space at ∞ of the self-adjoint relation in the minimal model (short for self-adjoint operator realization) of N contains a κ-dimensional non-positive subspace. In this paper we discuss two specific models for the function N ∈ N∞ κ: one associated with the irreducible representation of N and one associated with a regularized version of this representation which need not be irreducible. The state space in each of these models is a reproducing kernel Pontryagin space whose reproducing kernel is a matrix function constructed from the data in the representation.
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| 9. |
- Emmel, Christian, et al.
(författare)
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Realizations of meromorphic functions of bounded type
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Tidskriftsartikel (refereegranskat)abstract
- In this article it is shown that every function meromorphic in the upper halfplane and of bounded type does have a realization with the resolvent of a self-adjoint relation in a Kreinspace.
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| 10. |
- Emmel, Christian, 1995-
(författare)
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Realizations of meromorphic functions of bounded type
- 2023
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- A realization of a locally analytic function is basically a way of expressing the function in terms of the resolvent of a self-adjoint operator. In the classical Hilbert space setting realizations correspond in an essentially one-to-one way to Herglotz-Nevanlinna functions. Considering more generalobjects than Hilbert spaces, so called Krein spaces, opens up the possibility to address more general function classes. This thesis is concerned withrealizations and related minimality questions for meromorphic functionsof bounded type. In the general introduction we review preliminaries from complex anal-ysis and operator theory and give an idea how the main results of thesubsequent two articles relate to the already existing theory. In the first included article realizations for meromorphic functions ofbounded type are constructed. However, these realizations are not mini-mal in general. The second article is a first step in addressing this mini-mality question. There we focus on the well behaved subclass of atomic density functions for which minimal realizations are constructed.
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