Pencils of the form A(λ) = λE−A are studied, where A and E are bounded linear operators on a Hilbert space. Of interest are the spectral properties of A(λ). This is done via a corresponding linear relation in a Krein space, which is given in range representation using the two operators A and E. Under…
One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we obtain a new characterization for the subclass of non-negative…
We investigate the indefinite Kirchhoff Laplacian on a star graph G with n = n+ + n− edges of unit length, where the differential operator acts as minus second derivative on n_+ edges and plus second derivative on n_− edges. The operator is subject to Dirichlet conditions at the outer vertices and Kirchhoff…
The well-known Weyr characteristic for matrices, which counts the amount of linearly independent Jordan chains of a certain length, generalizes to linear relations in finite dimensional spaces. The Weyr characteristic of a linear relation consists of three finite sequences counting different types of…
We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schroedinger equation that is used to write the solution for quantum graphs. Formulation of the problem and derivation of secular equation for arbitrary quantum graphs is presented.…
We consider a quantum particle under the dynamical confinement caused by PT-symmetric box with a moving wall. The latter is described in terms of the time-dependent Schroedinger equation obeying the time-dependent PT-symmetric boundary conditions. The class of the functions, desscribing time-dependence…
We propose a general approach for deriving transparent boundary conditions for the stationary Schrödinger equation with arbitrary potential. It is proven that the transparent boundary conditions can be written in terms of the Weyl-Titchmarsh coefficients. As examples for the application of the proposed…
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.…
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…
We use the recently introduced Weyr characteristic of linear relations in C^n 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.
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…
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…
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…