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 linearly independent chains. Given a linear relation, the Weyr characteristics of its inverse, its adjoint, and its orthogonal complement are computed.

Locally Locally definitizable operators definitizable Spectrum operators have locally the same spectral properties as definitizable operators in Kreĭn spaces. It is shown in this note how to define spectral points of positive/negative type and spectral points of type $$\pi _{+}/\pi _{-}$$ via approximative eigensequences. This approach has the advantage that it does not make use of a local spectral function. Moreover, perturbation results for locally definitizable operators are discussed. Spectral points of type π + and π − are stable under compact perturbations. For real spectral points of type π + and type π − which are not in the interior of the spectrum the growth of the resolvent in an open neighborhood of these spectral points is of finite order. This can be utilized to show the existence of a local spectral function with singularities. With the help of this local spectral function one can also characterize spectral points of positive/negative type and spectral points of type π + and type π −: It turns out that all spectral subspaces corresponding to sufficiently small neighborhoods of spectral points of positive/negative type are Hilbert or anti-Hilbert spaces and spectral subspaces corresponding to spectral points of type π + or type π − are Pontryagin spaces. Locally definitizable operators are used in the study of indefinite Sturm–Liouville problems, λ-dependent boundary value problems, $$\mathcal{P}\mathcal{T}$$ -symmetric operators, and partial differential equations and in the study of problems of Klein–Gordon type.

The solvability of two-point boundary value problems for constant-coefficient differential-algebraic equations is investigated. Unlike previous studies, which often assume that the matrix pair associated with the equation is regular, we consider the general singular case. Using the Kronecker canonical form, we decompose the problem into simpler subsystems, enabling a systematic analysis of solvability. The method of parameterization reduces the boundary value problem to a system of algebraic equations. We derive criteria for the existence and uniqueness of solutions and provide a comprehensive framework for solving such boundary value problems. Several illustrative examples are presented and show the applicability of the results.

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 operators in Krein spaces which is based on local sign type properties of the spectrum and growth conditions on the resolvent. Based on these local properties, a notion of local non-negativity for self-adjoint operators in Krein spaces is defined and it is shown that such classes of operators appear naturally as perturbations of non-negative operators.

We study the behavior of eigenvalues of regular matrix pencils under rank-one perturbations which depend on a scalar parameter. In particular, 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, is described. Besides that, an interlacing result for rank-one perturbations of matrix pencils is obtained. Finally, we show how to use these results in the redesign of electrical circuits, like for the low pass filter or for a two-stage CMOS operational amplifier. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

On a star graph G with n = n_+ + n_- edges of unit length, we study the operator -d^2/d x^2 on n_+ and d^2/d x^2 on n_- edges equipped with Dirichlet boundary conditions at the outer vertices and a Kirchhoff condition at the central vertex. We study the spectral properties of the corresponding indefinite Kirchhoff Laplacian on G and we show that it is similar to a selfadjoint operator in the Hilbert space L^2(G) and that its eigenfunctions form a Riesz basis. Furthermore, we give a complete description of the point spectrum.

Given two linear relations Sand Tin Cn, we characterize when there exist linear relations equivalent to Sand T, respectively, such that one dimensional perturbations of each other; i.e., linear subspace of codimension at most one both in T, but it cannot be equal to both of them simultaneously. The result is achieved relating one dimensional perturbations of linear relations with rank-one perturbations of matrix pencils. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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 $\mathbb C$ which is motivated by the recent intensive research connected with $\mathcal P \mathcal T$-symmetric Hamiltonians.

We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrodinger 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. Application of the approach for the star graph is demonstrated by obtaining eigenfunctions and eigenvalues explicitely. Practical application of the model in conducting polymers and branched molecular chains is discussed.

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.

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 Schrödinger equation obeying the time-dependent PT-symmetric boundary conditions. The class of the functions, describing time-dependence of the wall's position and keeping the system as PT-symmetric is found. Physically observable characteristics, such as average kinetic energy and the average quantum force are calculated as a function of time. Also, geometric phase is calculated for the harmonically oscillating wall regime. Experimental realization of the proposed model is discussed.

We consider the indefinite Sturm–Liouville differential expression 𝔞(f):= - 1/w( 1/r f' ) ', where 𝔞 is defined on a finite or infinite open interval I with 0∈ I and the coefficients r and w are locally summable and such that r(x) and (sgn x) w(x) are positive a.e. on I. With the differential expression 𝔞 we associate a nonnegative self-adjoint operator A in the Krein space L^2_w(I) which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space L^2_|w|(I) . These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.

We apply the boundary triple technique to construct a coupling ( A , B ) (A,B) of two dual pairs ( A + , B + ) (A_+,B_+) and ( A − , B − ) (A_-,B_-) relative to some boundary triples. The notion of a real dual pair with respect to the time reversal operator \cT is introduced and it is shown that the coupling of two real dual pairs corresponding to real boundary triples is also real. If the operator \cP \cT intertwines the dual pairs ( A + , B + ) (A_+,B_+) and ( A − , B − ) (A_-,B_-) for some parity operator \cP , then it is shown that there exists a coupling ( A , B ) (A,B) of two dual pairs ( A + , B + ) (A_+,B_+) and ( A − , B − ) (A_-,B_-) such that the operator A A is \cP \cT -symmetric and \cP -symmetric in the Krein space (\sH ,<\cdot ,\cdot <) with the fundamental symmetry \cP . As the main result we describe proper extensions of A A which are \cP \cT -symmetric and \cP -selfadjoint. We apply this result to interpret the \cP \cT -symmetric Hamiltonian considered in Bender & Boettcher (1998) as a member of a family of \cP \cT -symmetric and \cP -selfadjoint extensions of the corresponding minimal operator. Keywords: dual pair; boundary triple; coupling; \cP \cT -symmetric operator; non-Hermitian Hamiltonian

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 Schrodinger 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.

. 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 x 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 x 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.

We develop relative oscillation theory for general Sturm–Liouville differential expressions of the form1r(−ddxpddx+q) and prove perturbation results and invariance of essential spectra in terms of the real coefficients p, q, r. The novelty here is that we also allow perturbations of the weight function r in which case the unperturbed and the perturbed operator act in different Hilbert spaces.

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.

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.

We study perturbations of the self-adjoint periodic Sturm–Liouville operatorA0=1r0(−ddxp0ddx+q0) and conclude under L1-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, 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.

In this note we provide estimates for the lower bound of the self-adjoint operator associated with the three-coefficient Sturm–Liouville differential expression 1 r ( − d d x p d d x + q ) \begin{equation*} \frac {1}{r} \left (-\frac {\mathrm d}{\mathrm dx} p \frac {\mathrm d}{\mathrm dx} + q\right ) \end{equation*} in the weighted L 2 L^2 -Hilbert space L 2 ( R ; r d x ) L^2(\mathbb R; rdx) .