Publikationen Prof. Trunk

Publikationen der Mitarbeiter

Publikationen am Fachgebiet

Results: 170
Created on: Thu, 28 Mar 2024 23:09:10 +0100 in 0.0766 sec


Azizov, Tomas Ya.; Trunk, Carsten;
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].



http://www.db-thueringen.de/servlets/DocumentServlet?id=24009
Behrndt, Jussi; Leben, Leslie; Martínez Pería, Francisco; Trunk, Carsten
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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=23864
Behrndt, Jussi; Möws, Roland; Trunk, Carsten
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
Ilchmann, Achim; Selig, Tilman; Selig, Tilman *1985-*; Trunk, Carsten;
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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=23009
Behrndt, Jussi; Leben, Leslie; Martínez Pería, Francisco; Möws, Roland; Trunk, Carsten
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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=22747
Trunk, Carsten;
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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=22710
Behrndt, Jussi; Philipp, Friedrich; Trunk, Carsten;
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
Möws, Roland;
Spektrallücken von indefiniten Sturm-Liouville-Operatoren, 2013. - Online-Ressource (PDF-Datei: XI, 67 S., 820,4 KB) : Ilmenau, Techn. Univ., Diss., 2013
Unterschiede zwischen dem gedruckten Dokument und der elektronischen Ressource können nicht ausgeschlossen werden

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.



http://www.db-thueringen.de/servlets/DocumentServlet?id=22550
Philipp, Friedrich; Trunk, Carsten;
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
Behrndt, Jussi; Luger, Annemarie; Trunk, Carsten
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

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