Indefinite Sturm-Liouville operators in polar form. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 96 (2024), 2, S. 1-58
https://doi.org/10.1007/s00020-023-02746-3
Feedback rectifiable pairs and stabilization of switched linear systems. - In: Systems & control letters, ISSN 1872-7956, Bd. 186 (2024), 105755, S. 1-10
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.
https://doi.org/10.1016/j.sysconle.2024.105755
The rank-one perturbation problem for linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (29 Seiten). - (Preprint ; M23,12)
We use the recently introduced Weyr characteristic of linear relations in Cn and its relation with the Kronecker canonical form of matrix pencils to describe their dimension. Then, this is applied to study one-dimensional perturbations of linear relations.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200305
On the essential spectrum of operator pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (9 Seiten). - (Preprint ; M23,11)
For a closed densely defined linear operator A and a bounded linear operator B on a Banach space X whose essential spectrums are contained in disjoint sectors, we show that the essential spectrum of the associated operator pencil λA + B is contained in a sector of the complex plane whose boundaries are determined purely by the angles that define the two sectors, which contain the essential spectrums of A and B.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200291
Lower bounds for self-adjoint Sturm-Liouville operators. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 151 (2023), 12, S. 5313-5323
https://doi.org/10.1090/proc/16523
Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (15 Seiten). - (Preprint ; M23,10)
The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in C which is motivated by the recent intensive research connected with PT-symmetric Hamiltonians.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200258
Functional models of symmetric and selfadjoint operators. - In: From complex analysis to operator theory: a panorama, (2023), S. 75-122
https://doi.org/10.1007/978-3-031-31139-0_7
Quantum particle under dynamical confinement: from quantum Fermi acceleration to high harmonic generation. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (15 Seiten). - (Preprint ; M23,09)
Quantum dynamics of a particle confined in a box with time-dependent wall is revisited by considering some unexplored aspects of the problem. In particular, the case of dynamical confinement in a time-dependent box in the presence of purely time-varying external potential is treated by obtaining exact solution. Also, some external potentials approving separation of space and time variables in the Schrödinger equation with time-dependent boundary conditions are classified. Time-dependence of the average kinetic energy and average quantum force are analyzed. A model for optical high harmonic generation in the presence of dynamical confinement and external linearly polarized monochromatic field is proposed.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200218
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (26 Seiten). - (Preprint ; M23,08)
We study singular Sturm-Liouville operators of the form 1/r_j (-d/dx p_j d/dx +q_j), j=0,1, in L_2((a; b); rj ), where, in contrast to the usual assumptions, the weight functions r_j have different signs near the singular endpoints a and b. In this situation the associated maximal operators become self-adjoint with respect to indefnite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r_j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L_1 perturbations of periodic operators.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200208
Feedback rectifiable pairs and stabilization of switched linear systems. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (12 Seiten). - (Preprint ; M23,07)
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop switched system is in upper triangular form. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. Several examples illustrate the characteristics of the problem considered and the application of the proposed constructive procedure.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200194
On the solvability of boundary value problems for linear differential-algebraic equations with constant coefficients. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (7 Seiten). - (Preprint ; M23,06)
We study a two-point boundary value problem for a linear differential-algebraic equation with constant coefficients by using the method of parameterization. The parameter is set as the value of the continuously differentiable component of the solution at the left endpoint of the interval. Applying the Weierstrass canonical form to the matrix pair associated with the differential-algebraic equation, we obtain a criterion for the unique solvability of the problem.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200182
Perturbations of periodic Sturm-Liouville operators. - In: Advances in mathematics, ISSN 1090-2082, Bd. 422 (2023), 109022, S. 1-22
https://doi.org/10.1016/j.aim.2023.109022
Indefinite Sturm-Liouville operators in polar form. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (49 Seiten). - (Preprint ; M23,05)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200114
PT-symmetric couplings of dual pairs. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (24 Seiten). - (Preprint ; M23,03)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200049
Spectral inclusion property for a class of block operator matrices. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (12 Seiten). - (Preprint ; M23,02)
The numerical range and the quadratic numerical range is used to study the spectrum of a class of block operator matrices. We show that the approximate point spectrum is contained in the closure of the quadratic numerical range. In particular, the spectral enclosures yield a spectral gap. It is shown that these spectral bounds are tighter than classical numerical range bounds.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200026
Lower bounds for self-adjoint Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (11 Seiten). - (Preprint ; M23,01)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200011
Relative oscillation theory and essential spectra of Sturm-Liouville operators. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 518 (2023), 1, 126673
https://doi.org/10.1016/j.jmaa.2022.126673
Bifurcations from codimension-one D4m-equivariant homoclinic cycles. - Ilmenau : Universitätsbibliothek, 2022. - 1 Online-Ressource (277 Seiten)
Technische Universität Ilmenau, Dissertation 2022
Das Thema dieser Arbeit ist eine detaillierte Beschreibung der Dynamik in der Nähe von D4m-symmetrischen relativen homoklinen Zykeln mit Hilfe von Lins Methode. Die homoklinen Zykel haben die Kodimension-1, d.h. wir beobachten ihre generische Entfaltung innerhalb einer einparametrigen Familie. Sie bestehen aus mehreren Trajektorien, die sowohl für positive als auch negative Zeit derselben hyperbolischen Gleichgewichtslage zustreben (Homokline Trajektorien) und die alle durch die von einer endlichen Gruppe induzierten Symmetrie voneinander abhängig sind. Wir nehmen reelle führende Eigenwerte und homokline Trajektorien an, die sich der Gleichgewichtslage entlang führender Richtungen nähern. Die Homoklinen befinden sich in flussinvarianten Unterräumen. Insbesondere für solche homoklinen Zykel in Differentialgleichungen mit Dk-Symmetrie (Dk ist die Symmetriegruppe eines regelmäßigen k-Ecks in der Ebene), bei denen k ein Vielfaches von 4 ist, stehen einige dieser flussinvarianten Unterräume senkrecht zueinander. Dies impliziert das Verschwinden der typischerweise auftretenden Terme führender exponentieller Konvergenzordnung in einigen der aus Lins Methode gewonnenen Bestimmungsgleichungen. Um eine genaue Beschreibung der nichtwandernden Dynamik eines solchen homoklinen Zykels zu geben, d.h. eine Beschreibung der Lösungen, die in der Umgebung des Zykels sowohl im Phasen- als auch im Parameterraum verbleiben, sind weitere Informationen über die Restterme in den Bestimmungsgleichungen erforderlich. In dieser Arbeit stellen wir eine verfeinerte Darstellung der Restterme in den Bestimmungsgleichungen vor und identifizieren zwei weitere Terme mit nächsthöheren exponentiellen Konvergenzraten. Darauf aufbauend diskutieren wir die Lösbarkeit der resultierenden Bestimmungsgleichungen für homokline Zykel in R4. Dabei sind zwei Fälle zu unterscheiden, die vom Größenverhältnis der beiden neuen Terme abhängen. In einem Fall beobachten wir einen endlichen Subshift. Im anderen Fall erweist sich die Analysis als schwieriger, so dass wir die Untersuchung auf periodische Lösungen beschränken.
https://doi.org/10.22032/dbt.55480
A Jordan-like decomposition for linear relations in finite-dimensional spaces. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (34 Seiten). - (Preprint ; M22,05)
A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space H (i.e., A is a linear subspace of H × H and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product H × H), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue infinity; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200249
Eigenvalues of parametric rank one perturbations of matrix pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (37 Seiten). - (Preprint ; M22,04)
The behavior of eigenvalues of regular matrix pencils under rank one perturbations which depend on a scalar parameter is studied. In particular we address the change of the algebraic multiplicities, the change of the eigenvalues for small parameter variations as well as the asymptotic eigenvalue behavior as the parameter tends to infinity. Besides that, an interlacing result for rank one perturbations of matrix pencils is obtained. Finally, we apply the result to a redesign problem for electrical circuits.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200237
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 295 (2022), 6, S. 1113-1162
The paper is a continuation of Part I and contains several further results on generalized boundary triples, the corresponding Weyl functions, and applications of this technique to ordinary and partial differential operators. We establish a connection between Post's theory of boundary pairs of closed nonnegative forms on the one hand and the theory of generalized boundary triples of nonnegative symmetric operators on the other hand. Applications to the Laplacian operator on bounded domains with smooth, Lipschitz, and even rough boundary, as well as to mixed boundary value problem for the Laplacian are given. Other applications concern with the momentum, Schrödinger, and Dirac operators with local point interactions. These operators demonstrate natural occurrence of ES$ES$-generalized boundary triples with domain invariant Weyl functions and essentially selfadjoint reference operators A0.
https://doi.org/10.1002/mana.202000049
Relative oscillation theory and essential spectra of Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (15 Seiten). - (Preprint ; M22,02)
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200169
On characteristic invariants of matrix pencils and linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (35 Seiten). - (Preprint ; M22,01)
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200140
The spectra of indefinite singular Sturm-Liouville operators. - Ilmenau : Universitätsbibliothek, 2021. - 1 Online-Ressource (92 Seiten)
Technische Universität Ilmenau, Dissertation 2021
In der vorliegenden Arbeit werden die spektralen Eigenschaften singulärer Sturm-Liouville-Differentialoperatoren der Form Af=1/r(-(pf')' + qf) mit reellwertigen Koeffizienten p, q und r untersucht. Hierbei betrachten wir indefinite Gewichtsfunktionen r. Basierend auf Erkenntnissen der relativen Oszillationstheorie sowie der Floquet-Theorie für periodische Sturm-Liouville-Operatoren werden Kriterien nachgewiesen, welche die Stabilität der essentiellen Spektren unter Störung der Koeffizienten sicherstellen. Außerdem wird die Häufung von Eigenwerten in den Lücken des essentiellen Spektrums untersucht. Wir formulieren Bedingungen, die eine Häufung der Eigenwerte innerhalb einer Lücke implizieren, bzw. eine Häufung ausschließen. Weiterhin werden die nichtreellen Spektren indefiniter Sturm-Liouville-Operatoren untersucht. Hierbei werden Schranken der nichtreellen Eigenwerte hinsichtlich ihres Absolutbetrages and Imaginärteils bestimmt. Der Nachweis der Schranken beruht auf einer gewissenhaften Analyse der zugehörigen Eigenfunktionen.
https://doi.org/10.22032/dbt.50260
PT-symmetric Hamiltonians as couplings of dual pairs. - In: Contributions to mathematics and statistics, (2021), S. 55-68
Spectral perturbation & optimization of matrix pencils. - Ilmenau : Universitätsverlag Ilmenau, 2021. - 1 Online-Ressource (130 Seiten)
Technische Universität Ilmenau, Dissertation 2021
In dieser Arbeit untersuchen wir lineare differentiell-algebraischen Gleichungen (DAEs). Die Lösungen solcher DAEs werden durch Eigenwerte und Hauptvektoren von Matrixbüscheln beschrieben. Hierdurch kann insbesondere das qualitative Verhalten der Lösungen durch eine gezielte Veränderung (oder Störung), hinsichtlich gewisser Robustheits- oder Stabilitätsvorgaben, verbessert werden. Wir untersuchen zunächst das Verhalten der Eigenwerte und Hauptvektoren von Matrixbüscheln unter Störungen niedrigen Ranges. Zur Beschreibung des Störverhaltens nutzen wir einen neuartigen Zugang mit linearen Relationen und einem Zusammenspiel der Segre und Weyr Charakteristiken. Von besonderem Interesse ist dabei das Problem der Eigenwertplatzierbarkeit durch Störungen niedrigen Ranges. Hierbei wird untersucht, ob eine vorgegebene Eigenwertlage durch eine gezielte Veränderung der DAE erreicht werden kann. Durch die Vorgabe der Eigenwertlage wird indirekt das Stabilitätsverhalten der DAE beeinflusst. Vereinfacht gesagt wird in dieser Arbeit gezeigt, dass jede vorgegebene Eigenwertlage durch eine Störung mit Rang eins realisierbar ist. Als Anwendung betrachten wir eine Designoptimierung von Operationsverstärkern, welche in den letzten Jahren in der Arbeitsgruppe um Ralf Sommer (TU Ilmenau & Institut für Mikroelektronik- und Mechatronik-Systeme) entwickelt wurde. Hierbei wurden gezielt Kapazitäten in die Verstärkerschaltung eingefügt, um ihr Übertragungsverhalten nach gewissen Vorgaben zu beeinflussen. Dabei entspricht jede neue Kapazität einer Störung der DAE vom Rang eins. In diesem Kontext sind die Platzierungsergebnisse jedoch nur bedingt geeignet. Hier treten zusätzliche Einschränkungen der erlaubten Modifikationen der DAE auf, da nur sehr wenige Störungen als Kapazitäten in der Schaltung realisiert werden können. Bei der Designoptimierung ist man zudem an kleinstmöglichen Veränderungen der DAE interessiert, um die Produktionskosten des Verstärkers zu minimieren. Daher untersuchen wir im zweiten Teil der Arbeit, wie platzierende Störungen mit kleinstmöglicher Norm sowie mit vorgegebener Struktur bestimmt werden können. Dieses Vorgehen bezeichnen wir als Spektrale Optimierung. Zur Bestimmung einer approximativen Lösung dieses Optimierungsproblems wurde ein Algorithmus entwickelt, welcher dann bei der Designoptimierung von zwei industriellen Verstärkerschaltungen eingesetzt wird.
https://doi.org/10.22032/dbt.49285
The spectrum and the Weyr characteristics of operator pencils and linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (18 Seiten). - (Preprint ; M21,05)
The relation between the spectra of operator pencils with unbounded coeficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200091
Locally finite extensions and Gesztesy-Šeba realizations for the Dirac operator on a metric graph. - In: Operator theory, (2021), S. 25-54
Perturbations of periodic Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (17 Seiten). - (Preprint ; M21,04)
We study perturbations of self-adjoint periodic Sturm-Liouville operators and conclude under L1-assumptions on the differences of the coeffcients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coeffcients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200075
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (54 Seiten). - (Preprint ; M21,03)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200058
PT-symmetric Hamiltonians as couplings of dual pairs. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (15 Seiten). - (Preprint ; M21,02)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200042
Finite rank perturbations of linear relations and matrix pencils. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 15 (2021), 2, 37, insges. 37 S.
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.
https://doi.org/10.1007/s11785-021-01082-x
Square roots of H-nonnegative matrices. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 621 (2021), S. 29-49
https://doi.org/10.1016/j.laa.2021.03.006
Unitary boundary pairs for isometric operators in Pontryagin spaces and generalized coresolvents. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 15 (2021), 2, 32, insges. 52 S.
https://doi.org/10.1007/s11785-020-01073-4
Linear relations and their singular chains. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (17 Seiten). - (Preprint ; M21,01)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200018
Functional models for entire symmetric operators in rigged de Branges Pontryagin spaces. - In: Journal of functional analysis, ISSN 1096-0783, Bd. 280 (2021), 2, 108776
The theory of operator extensions in rigged Pontryagin spaces is used to develop two functional models for closed symmetric entire operators S with finite deficiency indices (p,p) acting in a separable Pontryagin space K. In the first functional model it is shown that every such operator S is unitarily equivalent to the multiplication operator in a de Branges-Pontryagin space B(E) of p×1 vector valued entire functions. The second functional model is used to parametrize a class of compressed resolvents of extensions ÜÞS of S in terms of the range of a linear fractional transformation that is associated with the model. This approach is independent of the methods used by Krein and Langer to parameterize a related class of extensions.
https://doi.org/10.1016/j.jfa.2020.108776
Port-Hamiltonian formulation of nonlinear electrical circuits. - In: Journal of geometry and physics, Bd. 159 (2021), 103959, insges. 15 S.
We consider nonlinear electrical circuits for which we derive a port-Hamiltonian formulation. After recalling a framework for nonlinear port-Hamiltonian systems, we model each circuit component as an individual port-Hamiltonian system. The overall circuit model is then derived by considering a port-Hamiltonian interconnection of the components. We further compare this modeling approach with standard formulations of nonlinear electrical circuits.
https://doi.org/10.1016/j.geomphys.2020.103959
Invariance of the essential spectra of operator pencils. - In: Operator theory, operator algebras and their interactions with geometry and topology, (2020), S. 203-219
A linear relation approach to port-Hamiltonian differential-algebraic equations. - [Hamburg[ : [Fachbereich Mathematik, Universität Hamburg], 2020. - 1 Online-Ressource (31 Seiten). - ([Hamburger Beiträge zur Angewandten Mathematik] ; [2020, 16])Titel der monographischen Reihe und Veröffentlichungsangabe von der Homepage entnommen
http://epub.sub.uni-hamburg.de/epub/volltexte/2020/112509/
Square roots of H-nonnegative matrices. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2020. - 1 Online-Ressource (24 Seiten). - (Preprint ; M20,01)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2020200426
Gut vorbereitet in die erste Mathematikklausur : Aufgaben und Lösungen
2., aktualisierte Auflage. - München : Hanser, 2020. - 1 Online-Ressource (240 Seiten). - (Hanser eLibrary) ISBN 978-3-446-46615-9
Fit in einer Woche! Dieses Buch ist als Begleiter für die Vorbereitung auf Mathematikklausuren des ersten Universitätssemesters konzipiert. Die mehr als 100 klausurrelevanten Aufgaben und Lösungen sind so ausgewählt, dass eine intensive Vorbereitung etwa einer Woche bedarf. In jedem Abschnitt finden Sie eine breite Auswahl von Aufgaben, die in Klausuren zur Höheren Mathematik I oder Analysis I gestellt wurden. Dazu wird eine ausführliche und möglichst einfache Lösung formuliert, mit der das entsprechende Thema gleichzeitig wiederholt wird. Mit einer Zusammenfassung der wesentlichen mathematischen Zusammenhänge und Verfahren schließt jeder Abschnitt. Die behandelten Themen sind: - Grenzwerte - Reihen und Potenzreihen - Komplexe Zahlen - Eigenschaften von Funktionen - Differentation und Extremwerte - Taylorpolynom und Restgliedabschätzung - Integration, partielle Integration und Substitutionsregel - Partialbruchzerlegung und Integration rationaler Funktionen - Uneigentliche Integrale - Fourierreihen - Vollständige Induktion - Lineare Gleichungssysteme, Rang und Determinante - Lineare Abbildungen, Basen und Eigenwerte - Analytische Geometrie Die mathematischen Abschnitte werden durch drei komplette Beispielklausuren mit Lösungen abgerundet. Zusätzlich findet der Leser zwei Abschnitte mit praktischen und fundierten Hinweisen zur Prüfungsvorbereitung. Es eignet sich für Studierende der Ingenieurwissenschaften, der Wirtschaftswissenschaften, Biologie, Chemie und Informatik zur Prüfungsvorbereitung für Erstsemesterklausuren im Bereich Mathematik. Auf plus.hanser-fachbuch.de finden Sie zu diesem Titel kostenloses digitales Zusatzmaterial: die Kapitelzusammenfassungen und das Wichtigste für eine Klausur in Analysis I bzw. Höherer Mathematik I auf zwei Seiten
https://dx.doi.org/10.3139/9783446466159
Spectral enclosures for a class of block operator matrices. - In: Journal of functional analysis, ISSN 1096-0783, Volume 278 (2020), issue 10, 108455
https://doi.org/10.1016/j.jfa.2019.108455
Operatortheorie für PT-symmetrische Quantenmechanik. - Ilmenau : Universitätsbibliothek, 2019. - 1 Online-Ressource (88 Seiten)
Technische Universität Ilmenau, Dissertation 2019
Eine Verallgemeinerung der klassischen Quantenmechanik stammt von C. M. Bender und S. Boettcher welche alle Axiome der Quantenmechanik übernahmen, außer der Bedingung, dass der Hamiltonoperator Hermitesch ist. Sie fordern stattdessen, dass der Hamiltonoperator PT-symmetrisch ist. Hier sind P beziehungsweise T die Parität und die Zeitumkehr. Besonderes Augenmerk liegt auf den speziellen Hamiltonoperatoren $$H = p^2 - (iz)^{N+2}, z \in \Gamma$$ auf einer Kontur \Gamma und mit einer natürlichen Zahl N. In der vorliegenden Arbeit behandeln wir die Operatoren H, sowie Hamiltonoperatoren mit einem allgemeineren PT-symmetrischen Potential q, erklärt auf einer keilförmigen Kontur \Gamma. Das dazugehörige Eigenwertproblem hat nach einer Parametrisierung der Kontur die Gestalt $$e^{\mp 2i\phi}w''(x) + q_{\pm}(x)w(x) = \lambda w(x), x \in R_{\pm}.$$ Für das zu H gehörige Problem gilt q_{\pm}(x) = -(ix)^{N+2}e^{\pm(N+2)i\phi}. Dies sind Sturm-Liouville Differentialgleichung auf (-\infty, 0] und [0,\infty), welche wir mit operatortheoretischen Methoden behandeln. Wir geben, mittels WKB-Analysis ein Grenzpunktfallkriterium an und für das spezielle Potential aus H eine vollständige Klassifikation bezüglich der Weylschen Grenzpunkt-/Grenzkreisfall Alternative. Wir definieren die zu den obigen Differentialgleichungen gehörenden minimalen und maximalen Operatoren, welche zueinander adjungiert bezüglich der komplexen Konjugation sind. Diese Operatoren sind auf den reellen Halbachsen definiert und wir fügen diese zu dem minimalen und maximalen Operator auf der ganzen Achse zusammen, die wiederum zueinander adjungiert bezüglich des neuen inneren Produktes [\cdot, \cdot] := (P\cdot, \cdot) sind. Mithilfe einer Kopplungsbedingung G \in C^{2×2} in Null erhalten wir den Operator A_G, eine Einschränkung des maximalen Operators. Diese Bedingung besitzt Freiheitsgrade und wir geben Bedingungen an G an, sodass A_G PT-symmetrisch oder [\cdot, \cdot]-selbstadjungiert ist. Dafür konstruieren wir ein Randtripel. Außerdem berechnen wir die Weyl-Funktion und erhalten somit eine Bedingung für die Existenz und Lage der Eigenwerte von A_G. Mithilfe der WKB-Analysis untersuchen wir diese Bedingung und können Bereiche der komplexen Ebene ausschließen, in denen sich kein Spektrum befindet. Ferner besitzt A_G strukturell dieselben Spektraleigenschaften wie die entsprechenden Operatoren auf den Halbachsen.
https://www.db-thueringen.de/receive/dbt_mods_00040253
Locating the extremal entries of the Fiedler vector for rose trees. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 19 (2019), 1, e201900408, insges. 2 S.
https://doi.org/10.1002/pamm.201900408
The non-real spectrum of a singular indefinite Sturm-Liouville operator with regular left endpoint. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 19 (2019), 1, e201900133, insges. 2 S.
https://doi.org/10.1002/pamm.201900133
Operator-based approach to PT-symmetric problems on a wedge-shaped contour. - In: Quantum studies, ISSN 2196-5617, Bd. 6 (2019), 3, S. 315-333
https://doi.org/10.1007/s40509-019-00197-3
Invertibility of 2 × 2 operator matrices. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 292 (2019), 11, S. 2411-2426
https://doi.org/10.1002/mana.201800351
On a class of non-Hermitian matrices with positive definite Schur complements. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 147 (2019), 6, S. 2375-2388
https://doi.org/10.1090/proc/14412
The non-real spectrum of a singular indefinite Sturm-Liouville operator with regular left endpoint. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (3 Seiten). - (Preprint ; M19,05)
https://www.db-thueringen.de/receive/dbt_mods_00038524
Spectral bounds for indefinite singular Sturm-Liouville operators with uniformly locally integrable potentials. - In: Journal of differential equations, ISSN 1090-2732, Bd. 267 (2019), 1, S. 468-493
https://doi.org/10.1016/j.jde.2019.01.013
Spectral enclosures for a class of block operator matrices. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (23 Seiten). - (Preprint ; M19,04)
We prove new spectral enclosures for the non-real spectrum of a class of 2x2 block operator matrices with self-adjoint operators A and D on the diagonal and operators B and -B* as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to J-frame operators.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2019200198
Schur reduction of trees and extremal entries of the Fiedler vector. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 570 (2019), S. 93-122
https://doi.org/10.1016/j.laa.2019.02.008
Invariance of the essential spectra of operator pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (15 Seiten). - (Preprint ; M19,03)
The essential spectrum of operator pencils with bounded coefficients in a Hilbert space is studied. Sufficient conditions in terms of the operator coefficients of two pencils are derived which guarantee the same essential spectrum. This is done by exploiting a strong relation between an operator pencil and a specific linear subspace (linear relation).
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2019200141
Operator based approach to PT-symmetric problems on a wedge-shaped contour. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (23 Seiten). - (Preprint ; M19,02)
We consider a second-order differential equation -y''(z)-(iz)^{N+2}y(z)=\lambda y(z), z\in \Gamma with an eigenvalue parameter \lambda \in C. In PT quantum mechanics z runs through a complex contour \Gamma in C, which is in general not the real line nor a real half-line. Via a parametrization we map the problem back to the real line and obtain two differential equations on [0,\infty) and on (-\infty,0]. They are coupled in zero by boundary conditions and their potentials are not real-valued. The main result is a classification of this problem along the well-known limit-point/ limit-circle scheme for complex potentials introduced by A.R. Sims 60 years ago. Moreover, we associate operators to the two half-line problems and to the full axis problem and study their spectra.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2019200020
Spectral bounds for indefinite singular Sturm-Liouville operators with uniformly locally integrable potentials. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (26 Seiten). - (Preprint ; M19,01)
The non-real spectrum of a singular indefinite Sturm-Liouville operator A=1/r (-d/dx p d/dx+q) with a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1/p,q,r in L^1_loc(R)) of isolated eigenvalues with finite algebraic multiplicity which are symmetric with respect to the real line. In this paper bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A are proved for uniformly locally integrable potentials q and potentials $q in L^s(R) for some s in [1,\infty]. The bounds depend on the negative part of q, on the norm of 1/p and in an implicit way on the sign changes and zeros of the weight function.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2019200016
The gap distance to the set of singular matrix pencils. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 564 (2019), S. 28-57
https://doi.org/10.1016/j.laa.2018.11.020
Systems with strong damping and their spectra. - In: Mathematical methods in the applied sciences, ISSN 1099-1476, Bd. 41 (2018), 16, S. 6546-6573
https://doi.org/10.1002/mma.5166
Spectral bounds for singular indefinite Sturm-Liouville operators with L1-potentials. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 146 (2018), 9, S. 3935-3942
Im Titel ist "1" hochgestellt
https://doi.org/10.1090/proc/14059
On a class of non-Hermitian matrices with positive definite Schur complements. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (11 Seiten). - (Preprint ; M18,09)
Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second has a positive definite Schur complement with respect to its submatrix A. Additionally, we show that K can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces.
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2018200139
Spectrum of J-frame operators. - In: Opuscula mathematica, ISSN 2300-6919, Bd. 38 (2018), 5, S. 623-649
https://doi.org/10.7494/OpMath.2018.38.5.623
Finite rank perturbations of linear relations and singular matrix pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (25 Seiten). - (Preprint ; M18,08)
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare the number of Jordan chains of length at least n corresponding to some eigenvalue to each other. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behaviour is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.
https://www.db-thueringen.de/receive/dbt_mods_00034936
Locally finite extensions and Gesztesy-Šeba realizations for the Dirac operator on a metric graph. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (27 Seiten). - (Preprint ; M18,07)
We study extensions of direct sums of symmetric operators S=\oplus S_n where n run through the natural numbers. In general there is no natural boundary triplet associated even if there is one for every S_n^*. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of S_n^* and investigate the self-adjointness, the semi-boundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2018200090
The gap distance to the set of singular matrix pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2018. - 1 Online-Ressource (22 Seiten). - (Preprint ; M18,05)
We study matrix pencils sE-A using the associated linear subspace ker[A,-E]. The distance between subspaces is measured in terms of the gap metric. In particular, we investigate the gap distance of a regular matrix pencil to the set of singular pencils and provide upper and lower bounds for it. A relation to the distance to singularity in the Frobenius norm is provided.
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2018200051
Prevention of solidification cracking during pulsed laser beam welding. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 17 (2017), 1, S. 405-406
https://doi.org/10.1002/pamm.201710172
A new bound for the distance to singularity of a regular matrix pencil. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 17 (2017), 1, S. 863-864
https://doi.org/10.1002/pamm.201710399
A new method for network redesign via rank one updates. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 17 (2017), 1, S. 857-858
https://doi.org/10.1002/pamm.201710396
An estimate on the non-real spectrum of a singular indefinite Sturm-Liouville operator. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 17 (2017), 1, S. 859-860
https://doi.org/10.1002/pamm.201710397
Coupling of definitizable operators in Kre&bovko;in spaces. - In: Nanosistemy: fizika, chimija, matematika, ISSN 2220-8054, Bd. 8 (2017), 2, S. 166-179
https://doi.org/10.17586/2220-8054-2017-8-2-166-179
Spectral bounds for singular indefinite Sturm-Liouville operators with L1-potentials. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (7 Seiten). - (Preprint ; M17,12)Im Titel ist "1" hochgestellt
The spectrum of the singular indefinite Sturm-Liouville operator A=sgn(.) (-d^2/dx^2)+q with a real potential q in L^1(R)$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound l lambda | <= |q|_{L^1}^2 on the absolute values of the non-real eigenvalues lambda of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L^1-norm of q and its negative part q_-.
http://nbn-resolving.de/urn:nbn:de:gbv:ilm1-2017200509
An estimate on the non-real spectrum of a singular indefinite Sturm-Liouville operator. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (5 Seiten). - (Preprint ; M17,10)
It will be shown with the help of the Birman-Schwinger principle that the non-real spectrum of the singular indefinite Sturm-Liouville operator $\operatorname{sgn}(\cdot)(-\mathrm d^2/\mathrm d x^2 +q)$ with a real potential $q\in L^1\cap L^2$ is contained in a circle around the origin with radius $\|q\|_{L^1}^2$.
https://www.db-thueringen.de/receive/dbt_mods_00032787
New lower bound for the distance to singularity of regular matrix pencils. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (5 Seiten). - (Preprint ; M17,10)
For regular matrix pencils $\Ac(s)=sE-A$ the distance to the nearest singular pencil in the Frobenius norm of the coefficients is called the distance to singularity. We derive a new lower bound for this distance by using the spectral theory of tridiagonal Toeplitz matrices.
https://www.db-thueringen.de/receive/dbt_mods_00032786
Prevention of solidification cracking during pulsed laser beam welding. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (7 Seiten). - (Preprint ; M17,09)
We present a mathematical model to describe laser beam welding based on the heat equation. Since the material coeff cients depend on the temperature, this leads to a quasi-linear parabolic partial differential equation. It is our goal to prevent solidif cation cracking. We address this problem by means of optimal control. It is the intensity prof le of the laser beam which acts as the control function. The main challenge is the formulation of a suitable objective function. In particular, high velocities of the solidif cation interface need to be properly penalized in order to deal with and avoid cracking phenomena.
https://www.db-thueringen.de/receive/dbt_mods_00032771
A new method for network redesign via rank one updates. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (5 Seiten). - (Preprint ; M17,08)
We present a method to place the eigenvalues of an electrical network towards a prescribed set of complex numbers by inserting an additional capacitance into the network. We use recent results on rank one perturbations of regular matrix pencils and provide an upper bound on the approximation error of the eigenvalues in the chordal distance.
https://www.db-thueringen.de/receive/dbt_mods_00032770
Eigenvalue placement for regular matrix pencils with rank one perturbations. - In: SIAM journal on matrix analysis and applications, ISSN 1095-7162, Bd. 38 (2017), 1, S. 134-154
http://dx.doi.org/10.1137/16M1066877
Numerical range and quadratic numerical range for damped systems. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (27 Seiten). - (Preprint ; M17,05)
We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations z'' (t) + D z' (t) + A_0 z(t) = 0 in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as A_0. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients A_0 and D which improve earlier results for sectorial and selfadjoint D; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.
https://www.db-thueringen.de/receive/dbt_mods_00031984
Coupling of definitizable operators in Krein spaces. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (18 Seiten). - (Preprint ; M17,03)
Indefinite Sturm-Liouville operators defined on the real line are often considered as a coupling of two semibounded symmetric operators defined on the positive and the negative half axis. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on the real line.
https://www.db-thueringen.de/receive/dbt_mods_00031469
Spectrum of J-frame operators. - Ilmenau : Technische Universität, Institut für Mathematik, 2017. - 1 Online-Ressource (20 Seiten). - (Preprint ; M17,01)
A J-frame is a frame F for a Krein space which is compatible with the indefinite inner product in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in H . With every J-frame the so-called J-frame operator is associated, which is a self-adjoint operator in the Krein space H. The J-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of J-frame operators in a Krein space by a 2X2 block operator representation. The J-frame bounds of F are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the 2X2 block representation. Moreover, this 2X2 block representation is utilized to obtain enclosures for the spectrum of J-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all J-frames associated with a given J-frame operator.
https://www.db-thueringen.de/receive/dbt_mods_00031058
Eigenvalue estimates for operators with finitely many negative squares. - In: Opuscula mathematica, ISSN 2300-6919, Bd. 36 (2016), 6, S. 717-734
https://doi.org/10.7494/OpMath.2016.36.6.717
On stability of closedness and self-adjointness for 2 x 2 operator matrices. - In: Mathematical notes, ISSN 1573-8876, Bd. 100 (2016), 5, S. 870-875
http://dx.doi.org/10.1134/S0001434616110274
Variational principles for self-adjoint operator functions arising from second-order systems. - In: Operators and matrices, Bd. 10 (2016), 3, S. 501-531
http://dx.doi.org/10.7153/oam-10-29
Bounds on the non-real spectrum of a singular indefinite Sturm-Liouville operator on R. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 16 (2016), 1, S. 881-882
http://dx.doi.org/10.1002/pamm.201610429
On the parametric eigenvalue behavior of matrix pencils under rank one perturbations. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 16 (2016), 1, S. 873-874
http://dx.doi.org/10.1002/pamm.201610425
Limit-point/limit-circle classification of second-order differential operators arising in PT quantum mechanics. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 16 (2016), 1, S. 871-872
http://dx.doi.org/10.1002/pamm.201610424
Ob ustojčivosti zamknutosti i samosoprjažennosti dlja 2 x 2 operator-matric. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (6 Seiten). - (Preprint ; M16,07)
Consider an operator which is defined in Banach or Hilbert space by a 2x2 matrix with entries A, B, C, D which where linear operators and which are assumed to be unbounded. In the case when the operators C and B are relatively bounded with respect to the operators A and D, respectively, new conditions of the closeness or closability are obtained for the operator L. For the operator L acting in a Hilbert space the analogs of Rellich-Kato theorems on the stability of self-adjointness are obtained.
https://www.db-thueringen.de/receive/dbt_mods_00030580
Non-negative operators in Krein spaces and rank one perturbations. - Ilmenau : Universitätsverlag Ilmenau, 2016. - Online-Ressource (116 Seiten, 6.65 MB)
Technische Universität Ilmenau, Dissertation 2016
In der vorliegenden Arbeit werden eindimensionale Störungen von nichtnegativen Operatoren in Kreinräumen betrachtet. Dabei wird untersucht wie sich die Anzahl der Eigenwerte und deren Vielfachheit in einer Lücke des essentiellen Spektrums unter einer Störung ändern können. Zudem wird beschrieben wie sich an einem Eigenwert die Anzahl und die Länge der linear unabhängigen Jordanketten ändern können.
https://www.db-thueringen.de/receive/dbt_mods_00029981
Limit-point/limit-circle classification of second-order differential operators arising in PT quantum mechanics. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (6 Seiten). - (Preprint ; M16,03)
https://www.db-thueringen.de/receive/dbt_mods_00029270
Bounds on the non-real spectrum of a singular indefinite Sturm-Liouville operator on R. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (4 Seiten). - (Preprint ; M16,05)
A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(&hahog;) and an integrable, continuous potential q is obtained.
https://www.db-thueringen.de/receive/dbt_mods_00029271
On the parametric eigenvalue behavior of matrix pencils under rank one perturbations. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (8 Seiten). - (Preprint ; M16,04)
We study the eigenvalues of rank one perturbations of regular matrix pencils depending linearly on a complex parameter. We prove properties of the corresponding eigenvalue sets including a convergence result as the parameter tends to infinity and an eigenvalue interlacing property for real valued pencils having real eigenvalues only.
https://www.db-thueringen.de/receive/dbt_mods_00029233
The Byrnes-Isidori form for infinite-dimensional systems. - In: SIAM journal on control and optimization, ISSN 1095-7138, Bd. 54 (2016), 3, S. 1504-1534
http://dx.doi.org/10.1137/130942413
Eigenvalue estimates for operators with finitely many negative squares. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (14 Seiten). - (Preprint ; M16,02)
Let A and B be selfadjoint operators in a Krein space. Assume that the re- solvent difference of A and B is of rank one and that the spectrum of A consists in some interval I of isolated eigenvalues only. In the case that A is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of B in the interval I. The general results are applied to singular indefinite Sturm-Liouville problems.
https://www.db-thueringen.de/receive/dbt_mods_00029046
Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 439 (2016), 2, S. 864-895
http://dx.doi.org/10.1016/j.jmaa.2016.03.012
Eigenvalue placement for regular matrix pencils with rank one perturbations. - Ilmenau : Technische Universität, Institut für Mathematik, 2016. - 1 Online-Ressource (15 Seiten). - (Preprint ; M16,01)
A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in \C\cup\{\infty\} which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of sE-A may disappear and the sum of the length of all destroyed Jordan chains is the number of eigenvalues (counted with multiplicities) which can be placed arbitrarily in \C\cup\{\infty\}. We prove sharp upper and lower bounds of the change of the algebraic and geometric multiplicity of an eigenvalue under rank one perturbations. Finally we apply our results to a pole placement problem for a single-input differential algebraic equation with feedback.
http://www.db-thueringen.de/servlets/DocumentServlet?id=27311
Linear relations and the Kronecker canonical form. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 488 (2016), S. 13-44
http://dx.doi.org/10.1016/j.laa.2015.09.033
Spectral points of type π+ and type π- of closed operators in indefinite inner product spaces. - In: Operators and matrices, Bd. 9 (2015), 3, S. 481-506
Im Titel ist "+" und "-" tiefgestellt
http://dx.doi.org/10.7153/oam-09-30
Locally definitizable operators: the local structure of the spectrum. - In: Operator theory, (2015), S. 241-259
Linear relations and the Kronecker canonical form. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2015. - Online-Ressource (PDF-Datei: 27 S., 402 KB). - (Preprint ; M15,05)
We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents. These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part. Or, to state it more concise: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices.
http://www.db-thueringen.de/servlets/DocumentServlet?id=26272
The effect of finite rank perturbations on Jordan chains of linear operators. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 479 (2015), S. 118-130
http://dx.doi.org/10.1016/j.laa.2015.04.007
On a class of Sturm-Liouville operators which are connected to PT symmetric problems. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 14 (2014), 1, S. 991-992
http://dx.doi.org/10.1002/pamm.201410476
The invertibility of 2 x 2 operator matrices. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 19 S., 305 KB). - (Preprint ; M14,10)
In this paper the properties of right invertible row operators, i.e., of 1x2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2x2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.
http://www.db-thueringen.de/servlets/DocumentServlet?id=25047
Variational principles for self-adjoint operator functions arising from second-order systems. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 30 S., 455 KB). - (Preprint ; M14,09)
http://www.db-thueringen.de/servlets/DocumentServlet?id=25046
Spectral points of type π + and type π - of closed operators in indefinite inner product spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 27 S., 218 KB). - (Preprint ; M14,04)
We introduce the notion of spectral points of type π+ and type π- of closed operators A in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that A is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of A is of finite order in a neighborhood of a real spectral point of type π+ or π- which is not in the interior of the spectrum of A. Finally, we prove that there exists a local spectral function on intervals of type π+ or π-.
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On limit point and limit circle classification for PT symmetric operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 5 S., 103,3 KB). - (Preprint ; M14,03)
A prominent class of PT-symmetric Hamiltonians is $H:= 1/2 p^2 + x^2 (ix)^N, for x \in \Gamma$ for some nonnegative number N. The associated eigenvalue problem is defined on a contour $\Gamma$ in a specific area in the complex plane (Stokes wedges), see [3,5]. In this short note we consider the case N=2 only. Here we elaborate the relationship between Stokes lines and Stokes wedges and well-known limit point/limit circle criteria from [11, 6, 10].
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The effect of finite rank perturbations on Jordan chains of linear operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2014. - Online-Ressource (PDF-Datei: 12 S., 280,7 KB). - (Preprint ; M14,02)
A general result on the structure and dimension of the root subspaces of a matrix or a linear operator under finite rank perturbations is proved: The increase of dimension from the n-th power of the kernel of the perturbed operator to the (n+1)-th power differs from the increase of dimension of the corresponding powers of the kernels of the unperturbed operator by at most the rank of the perturbation and this bound is sharp.
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On finite rank perturbations of selfadjoint operators in Krein spaces and eigenvalues in spectral gaps. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 8 (2014), 4, S. 925-936
https://doi.org/10.1007/s11785-013-0318-2
The Byrnes-Isidori form for infinite-dimensional systems. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 28 S., 470,8 KB). - (Preprint ; M13,14)
We define a Byrnes-Isidori form for a class of infinite-dimensional systems with relative degree r and show that any system belonging to this class can be transformed into this form. We also analyze the concept of (stable) zero dynamics and show that it is, together with the Byrnes-Isidori form, instrumental for static proportional high-gain output feedback stabilization. Moreover, we show that funnel control is feasible for any system with relative degree one and with exponentially stable zero dynamics; a funnel controller is a time-varying proportional output feedback controller which ensures, for a large class of reference signals, that the error between the output and the reference signal evolves within a prespecified funnel. Therefore transient behavior of the error is obeyed.
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Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 32 S., 308 KB). - (Preprint ; M13,13)
Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I is an open interval such that the spectrum of A in I consists of isolated eigenvalues we prove sharp estimates on the numbers and multiplicities of eigenvalues of B in I. The general result is illustrated with eigenvalue estimates for singular left definite Sturm-Liouville differential operators.
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Locally definitizable operators: the local structure of the spectrum. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 16 S., 231 KB). - (Preprint ; M13,12)
We consider different types of spectral points of locally definitizable operators which can be defined with the help of approximate eigensequences. Their behavior allow a characterization in terms of the (local) spectral function. Moreover, we review some perturbation results for locally definitizable operators.
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Bounds on the non-real spectrum of differential operators with indefinite weights. - In: Mathematische Annalen, ISSN 1432-1807, Bd. 357 (2013), 1, S. 185-213
http://dx.doi.org/10.1007/s00208-013-0904-7
Spektrallücken von indefiniten Sturm-Liouville-Operatoren, 2013. - Online-Ressource (PDF-Datei: XI, 67 S., 820,4 KB) : Ilmenau, Techn. Univ., Diss., 2013
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In der Dissertationsschrift "Spektrallücken von indefiniten Sturm-Liouville-Operatoren" werden verschiedene Klassen von selbstadjungierten Operatoren und Relationen in indefiniten Innenprodukträumen betrachtet. Die Arbeit enthält zwei Hauptergebnisse: (A) Für lokal definisierbare Relationen wird gezeigt, dass die Endlichkeit der Anzahl der Eigenwerte in einer reellen Spektrallücke des essentiellen Spektrums unter endlichdimensionalen Störungen erhalten bleibt. (B) Für eine Unterklasse der lokal definisierbare Relationen, nämlich für Relationen mit endlich vielen negativen Quadraten, werden die Anzahl der Eigenwerte der gestörten Relation in einer reellen Spektrallücke des essentiellen Spektrums nach oben/unten durch die Anzahl der Eigenwerte derungestörten Relation und weiteren Korrekturgrößen abgeschätzt. Dabei werden hier nur eindimensionale Störungen betrachtet. Zudem gelingt der Nachweis, dass die in dieser Promotionsschrift vorgestellten Abschätzungen scharf sind.Diese abstrakten Ergebnisse aus dem ersten Teil der Arbeit werden im zweiten Teil auf Sturm-Liouville-Differentialoperatoren mit einer indefinitenGewichtsfunktion angewandt. In vielen Fällen werden die Abschätzungen leicht verbessert.
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The numerical range of non-negative operators in Krein spaces. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 438 (2013), 5, S. 2542-2556
We define and characterize the Krein space numerical range $W(A)$ and the Krein space co-numerical range $W_{\rm co}(A)$ of a non-negative operator $A$ in a Krein space. It is shown that the non-zero spectrum of $A$ is contained in the closure of $W(A)\cap W_{\rm co}(A)$.
http://dx.doi.org/10.1016/j.laa.2012.10.048
On the negative squares of a class of self-adjoint extensions in Krein spaces. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 286 (2013), 2/3, S. 118-148
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 $\mathcal D_\kappa^{n\times n}$-functions is introduced and, in particular, the properties of the inverse of a certain $\mathcal D_\kappa^{2\times 2}$-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.
http://dx.doi.org/10.1002/mana.201000154
Local spectral theory for normal operators in Krein spaces. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 286 (2013), 1, S. 42-58
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.
http://dx.doi.org/10.1002/mana.201000141
Some Sobelev spaces as Pontryagin spaces. - In: Vestnik Južno-Ural'skogo Gosudarstvennogo Universiteta. Serija matematika, mechanika, fizika / Južno-Uralьskij gosudarstvennyj universitet. - Čeljabinsk, 2014- , ISSN: 2075-809X , ZDB-ID: 2701282-7, ISSN 2075-809X, Bd. 6.2012, 11 (270), S. 14-23
We show that well known Sobolev spaces can quite naturally be treated as Pontryagin spaces. This point of view gives a possibility to obtain new properties for some traditional objects such as simplest differential operators.
On similarity of indefinite Sturm-Liouville operators. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 12 (2012), 1, S. 771-772
http://dx.doi.org/10.1002/pamm.201210374
On finite rank perturbations of selfadjoint operators in Krein spaces and eigenvalues in spectral gaps. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 10 S., 150,6 KB). - (Preprint ; M12,12)
It is shown that the finiteness of eigenvalues in a spectral gap of a definitizable or locally definitizable selfadjoint operator in a Krein space is preserved under finite rank perturbations. This results is applied to a class of singular Sturm-Liouville operators with an indefinite weight function.
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The numerical range of non-negative operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 21 S., 177,7 KB). - (Preprint ; M12,11)
We define and characterize the Krein space numerical range W(A) and the Krein space co-numerical range W_{\rm co}(A) of a non-negative operator A in a Krein space. It is shown that the non-zero spectrum of A is contained in the closure of W(A)\cap W_{\rm co}(A).
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Bounds on the non-real spectrum of differential operators with indefinite weights. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2012. - Online-Ressource (PDF-Datei: 27 S., 276,8 KB). - (Preprint ; M12,07)
Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and infinity are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm-Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.
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PT symmetric, Hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics. - In: Journal of mathematical physics, ISSN 1089-7658, Bd. 53 (2012), 1, S. 012109-1-012109-18
https://doi.org/10.1063/1.3677368
Spectral theory, mathematical system theory, evolution equations, differential and difference equations : 21st International Workshop on Operator Theory and Applications, Berlin, July 2010. - Basel : Birkhäuser, 2012. - VIII, 690 S.. - (Operator theory ; 221) ISBN 3-0348-0296-X
Literaturangaben
Local definitizability of T [*]T and TT[*]. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 71 (2011), 4, S. 491-508
http://dx.doi.org/10.1007/s00020-011-1913-0
Eigenvalue estimates for singular left-definite Sturm-Liouville operators. - In: Journal of spectral theory, ISSN 1664-0403, Bd. 1 (2011), 3, S. 327-347
The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the $J$-selfadjoint operator $JA$ is real and it follows that an interval $(a,b)\subset\dR^+$ is a gap in the essential spectrum of $A$ if and only if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of the $J$-selfadjoint operator $JA$. As one of the main results it is shown that the number of eigenvalues of $JA$ in $(-b,-a) \cup (a,b)$ differs at most by three of the number of eigenvalues of $A$ in the gap $(a,b)$; as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results.
https://doi.org/10.4171/JST/14
Selfadjoint operators in S-spaces. - In: Journal of functional analysis, ISSN 1096-0783, Bd. 260 (2011), 4, S. 1045-1059
We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by $\Skindef := (U\,\cdot\,,-)$, where $U$ is a unitary operator. We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator $A$ in an S-space we find an inner product which turns $\bez$ into a Krein space and $A$ into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well know and in general unsolved problem.
https://doi.org/10.1016/j.jfa.2010.10.023
Spectral points of definite type and type π for linear operators and relations in Krein spaces. - In: Journal of the London Mathematical Society, ISSN 1469-7750, Bd. 83 (2011), 3, S. 768-788
Spectral points of positive and negative type, and type $\pi_{+}$ and type $\pi_{-}$ for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric. Many of the obtained perturbation results are also new for the special case of bounded and unbounded selfadjoint operators in Krein spaces.
http://dx.doi.org/10.1112/jlms/jdq098
Frequency compensation for a class of DAE's arising in electrical circuits. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 11 (2011), 1, S. 837-838
Structured perturbations of regular pencils of the form $sE-A$, $E,A\in\dR^{n\times n}, ˜s\in\dC,$ are considered which model the addition of a capacitance $c$ in an electrical circuit in order to improve the frequency response.
http://dx.doi.org/10.1002/pamm.201110407
On a class of J-self-adjoint operators with empty resolvent set. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 379 (2011), 1, S. 272-289
https://doi.org/10.1016/j.jmaa.2010.12.048
PT symmetric, hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 28 S., 189,9 KB). - (Preprint ; M11,15)
In the recent years a generalization H=p^2 + x^2(ix)^\epsilon of the harmonic oscillator using a complex deformation was investigated, where \epsilon is a real parameter. Here, we will consider the most simple case: \epsilon even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.
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Frequency compensation for a class of DAE's arising in electrical circuits. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 5 S., 115,2 KB). - (Preprint ; M11,11)
http://www.db-thueringen.de/servlets/DocumentServlet?id=18489
A perturbation approach to differential operators with indefinite weights. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 18 S., 210,5 KB). - (Preprint ; M11,08)
http://www.db-thueringen.de/servlets/DocumentServlet?id=18137
Eigenvalue estimates for singular left-definite Sturm-Liouville operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2011. - Online-Ressource (PDF-Datei: 15 S., 379,8 KB). - (Preprint ; M11,01)
The spectral properties of a singular left-definite Sturm-Liouville operator JA are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart A which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the J-selfadjoint operator JA is real and it follows that an interval (a; b) \subset {\Bbb R}+ is a gap in the essential spectrum of A if and only if both intervals (-b;-a) and (a; b) are gaps in the essential spectrum of the J-selfadjoint operator JA. As one of the main results it is shown that the number of eigenvalues of JA in (-b;-a) [ (a; b) di ers at most by three of the number of eigenvalues of A in the gap (a; b); as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results
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Small perturbations of selfadjoint and unitary operators in Krein spaces. - In: Journal of operator theory, ISSN 0379-4024, Bd. 64 (2010), 2, S. 401-416
We investigate the behaviour of the spectrum of selfadjoint operators in Krein spaces under perturbations with uniformly dissipative operators. Moreover we consider the closely related problem of the perturbation of unitary operators with uniformly bi-expansive. The obtained perturbation results give a new characterization of spectral points of positive type and of type $\pi_{+}$ of selfadjoint (resp.\ unitary) operators in Krein spaces.
Hyponormal and strongly hyponormal matrices in inner product spaces. - In: Linear algebra and its applications, ISSN 0024-3795, Bd. 433 (2010), 6, S. 1055-1076
http://dx.doi.org/10.1016/j.laa.2010.04.050
On the negative squares of a class of self-adjoint extensions in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 34 S., 386 KB). - (Preprint ; M10,16)
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 t (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 n×n -functions is introduced and, in particular, the properties of the inverse of a certain D k 2×2 -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.
http://www.db-thueringen.de/servlets/DocumentServlet?id=17238
On a class of J-self-adjoint operators with empty resolvent set. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 29 S., 257,3 KB). - (Preprint ; M10,11)
http://www.db-thueringen.de/servlets/DocumentServlet?id=16642
Local spectral theory for normal operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2010. - Online-Ressource (PDF-Datei: 18 S., 172 KB). - (Preprint ; M10,09)
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of nite order, then the normal operator possesses a local spectral function dened for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space.
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Spectral theory for second order systems and indefinite Sturm-Liouville problems. - Getr. Zählung Ilmenau : Techn. Univ., Habil.-Schr., 2010
On domains of PT symmetric operators related to -y"(x) + (-1) n x 2n y(x). - In: Journal of physics. Mathematical and theoretical. - Bristol : IOP Publ., 2007- , ISSN: 1751-8121 , ZDB-ID: 1363010-6, ISSN 1751-8121, Bd. 43.2010, 17, 175303, insges. 13 S.
http://dx.doi.org/10.1088/1751-8113/43/17/175303
Spectral analysis of singular ordinary differential operators with indefinite weights. - In: Journal of differential equations, ISSN 1090-2732, Bd. 248 (2010), 8, S. 2015-2037
http://dx.doi.org/10.1016/j.jde.2009.11.026
Analyticity of semigroups related to a class of block operator matrices. - In: Operator algebras, operator theory and applications, (2010), S. 257-271
Recent advances in operator theory in Hilbert and Krein spaces : [7th Workshop on Operator Theory in Krein Spaces and Spectral Analysis, which was held at the Technische Universität Berlin, Germany, December 13 to 16, 2007]. - Basel : Birkhäuser, 2010. - XV, 307 S.. - (Operator theory ; 198) ISBN 978-3-0346-0179-5
Literaturangaben
Non-real eigenvalues of singular indefinite Sturm-Liouville operators. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 137 (2009), 11, S. 3797-3806
https://doi.org/10.1090/S0002-9939-09-09964-X
Definitizability of a class of J-selfadjoint operators with applications. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 9 (2009), 1, S. 665-666
http://dx.doi.org/10.1002/pamm.200910302
Singular indefinite Sturm-Liouville operators with a spectral gap. - In: Proceedings in applied mathematics and mechanics, ISSN 1617-7061, Bd. 9 (2009), 1, S. 669-670
http://dx.doi.org/10.1002/pamm.200910304
On domains of PT symmetric operators related to -y"(x) + (-1) n x 2n y(x). - Ilmenau : Techn. Univ., Inst. für Mathematik, 2009. - Online-Ressource (PDF-Datei: 15 S., 186,1 KB). - (Preprint ; M09,30)
http://www.db-thueringen.de/servlets/DocumentServlet?id=14408
Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics. - In: Semigroup forum, ISSN 1432-2137, Bd. 79 (2009), 1, S. 79-100
http://dx.doi.org/10.1007/s00233-009-9148-y
Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x. - In: Proceedings. Mathematics / The Royal Society of Edinburgh. - Cambridge [u.a.] : Cambridge Univ. Press, 1943- , ISSN: 1473-7124 , ZDB-ID: 2036780-6, ISSN 1473-7124, Bd. 139 (2009), 3, S. 483-503
http://dx.doi.org/10.1017/S0308210507000686
Compact and finite rank perturbations of closed linear operators and relations in Hilbert spaces. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 63 (2009), 2, S. 151-163
http://dx.doi.org/10.1007/s00020-008-1650-1
G-self-adjoint operators in Almost Pontryagin spaces. - In: Spectral theory in inner product spaces and applications, (2009), S. 207-235
On domains of powers of linear operators and finite rank perturbations. - In: Spectral theory in inner product spaces and applications, (2009), S. 31-36
Spectral theory in inner product spaces and applications : 6th Workshop on Operator Theory in Krein Spaces and Operator Polynomials, Berlin, December 2006. - Basel : Birkhäuser, 2009. - XX, 250 S.. - (Operator theory ; 188) ISBN 3-7643-8910-9
Literaturangaben
Analyticity and Riesz basis property of semigroups associated to damped vibrations. - In: Journal of evolution equations, ISSN 1424-3202, Bd. 8 (2008), 2, S. 263-281
http://dx.doi.org/10.1007/s00028-007-0351-6
Spectralizable operators. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 61 (2008), 3, S. 413-422
http://dx.doi.org/10.1007/s00020-008-1585-6
Spectral theory for operator matrices related to models in mechanics. - In: Mathematical notes, ISSN 1573-8876, Bd. 83 (2008), 5/6, S. 843-850
http://dx.doi.org/10.1134/S0001434608050295
Non-real eigenvalues of singular indefinite Sturm-Liouville operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - Online-Ressource (PDF-Datei: 10 S., 250,5 KB). - (Preprint ; M08,28)
http://www.db-thueringen.de/servlets/DocumentServlet?id=11843
Accumulation of complex eigenvalues of indefinite Sturm-Liouville operators. - In: Journal of physics, ISSN 1751-8121, Bd. 41.2008, 24, 244003, insges. 10 S.
http://dx.doi.org/10.1088/1751-8113/41/24/244003
Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - Online-Ressource (PDF-Datei: 20 S., 691 KB). - (Preprint ; M08,19)
http://www.db-thueringen.de/servets/DocumentServlet?id=11137
Analyticity of semigroups related to a class of block operator matrices. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - 15 S. = 245,8 KB, Text. - (Preprint ; M08,15)
http://www.db-thueringen.de/servlets/DocumentServlet?id=10740
Spectral points of definite type and type π for linear operators and relations in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - 29 S. = 311,6 KB, Text. - (Preprint ; M08,12)
http://www.db-thueringen.de/servlets/DocumentServlet?id=10405
G-selfadjoint operators in Almost Pontryagin spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - 29 S. = 330,1 KB, Text. - (Preprint ; M08,08)
http://www.db-thueringen.de/servlets/DocumentServlet?id=10383
Small perturbation of selfadjoint and unitary operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2008. - 12 S. = 246,6 KB, Text. - (Preprint ; M08,03)
http://www.db-thueringen.de/servlets/DocumentServlet?id=9918
On finite rank perturbations of definitizable operators. - In: Journal of mathematical analysis and applications, ISSN 1096-0813, Bd. 339 (2008), 2, S. 1161-1168
https://doi.org/10.1016/j.jmaa.2007.07.016
Minimum-phase infinite-dimensional second-order systems. - In: IEEE transactions on automatic control, ISSN 1558-2523, Bd. 52 (2007), 9, S. 1654-1665
http://dx.doi.org/10.1109/TAC.2007.904471
A Sturm-Liouville problem depending rationally on the eigenvalue parameter. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 280 (2007), 15, S. 1709-1726
http://dx.doi.org/10.1002/mana.200510573
Compact and finite rank perturbations of linear relations in Hilbert spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 13 S. = 248,7 KB, Text. - (Preprint ; M07,25)
http://www.db-thueringen.de/servlets/DocumentServlet?id=9797
Spectralizable operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 10 S. = 285,5 KB, Text. - (Preprint ; M07,24)
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Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 20 S. = 326,2 KB, Text. - (Preprint ; M07,23)
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Accumulation of complex eigenvalues of indefinite Sturm-Liouville operators. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 12 S. = 272,9 KB, Text. - (Preprint ; M07,22)
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On domains of powers of linear operators and finite rank perturbations. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 6 S. = 141,65 KB, Text. - (Preprint ; M07,21)
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Spectral theory for operator matrices related to models in mechanics. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 12 S. = 203,3 KB, Text. - (Preprint ; M07,20)
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Analyticity and Riesz basis property of semigroups associated to damped vibrations. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 20 S. = 285,5 KB, Text. - (Preprint ; M07,19)
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Perturbation theory for self-adjoint operators in Krein spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 12 S. = 206,6 KB, Text. - (Preprint ; M07,18)
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Some Sobolev spaces as Pontryagin spaces. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2007. - 14 S. = 239,8 KB, Text. - (Preprint ; M07,17)
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