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Prof. Dr. Achim Ilchmann

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Berger, Thomas; Giribet, Juan; Martínez Pería, Francisco; Trunk, Carsten;
On a class of non-Hermitian matrices with positive definite Schur complements. - In: Proceedings of the American Mathematical Society - Providence, RI : Soc, ISSN 10886826, Bd. 147 (2019), 6, S. 2375-2388

Behrndt, Jussi; Schmitz, Philipp; Trunk, Carsten;
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)
Behrndt, Jussi; Schmitz, Philipp; Trunk, Carsten;
Spectral bounds for indefinite singular Sturm-Liouville operators with uniformly locally integrable potentials. - In: Journal of differential equations - Orlando, Fla. : Elsevier, ISSN 1090-2732, Bd. 267 (2019), 1, S. 468-493

Giribet, Juan; Langer, Matthias; Martínez Pería, Francisco; Philipp, Friedrich; Trunk, Carsten;
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
Ilchmann, Achim; Leben, Leslie; Witschel, Jonas; Worthmann, Karl;
Optimal control of differential-algebraic equations from an ordinary differential equation perspective. - In: Optimal control, applications and methods - New York, NY [u.a.] : Wiley, ISSN 1099-1514, Bd. 40 (2019), 2, S. 351-366

https://doi.org/10.1002/oca.2481
Gernandt, Hannes; Pade, Jan Philipp;
Schur reduction of trees and extremal entries of the Fiedler vector. - In: Linear algebra and its applications : LAA. - New York, NY : American Elsevier Publ., Bd. 570 (2019), S. 93-122

Gernandt, Hannes; Moalla, Nedra; Philipp, Friedrich; Selmi, Wafa; Trunk, Carsten;
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
Leben, Florian; Trunk, Carsten;
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
Behrndt, Jussi; Schmitz, Philipp; Trunk, Carsten;
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
Berger, Thomas; Gernandt, Hannes; Trunk, Carsten; Winkler, Henrik; Wojtylak, Michał;
The gap distance to the set of singular matrix pencils. - In: Linear algebra and its applications : LAA. - New York, NY : American Elsevier Publ., Bd. 564 (2019), S. 28-57

https://doi.org/10.1016/j.laa.2018.11.020