Jordan Chains Research Articles

Jordan blocks and the Bethe Ansatz II: The eclectic spin chain beyond K = 1

We continue the classification of the Jordan chains of the eclectic three state spin chain that we started in our previous article. Following the same steps, we construct the generalised eigenvectors of this spin chain by computing the strongly twisted limit of linear combinations of eigenvectors of a twisted XXX SU(3) spin chain. We show that this classification problem can be mapped to the computation of the number of positive integer solutions of a system of linear equations.

Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

Jordan chains of h-cyclic matrices, II

McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145-159] gave a necessary condition on the structure of the Jordan chains of h-cyclic matrices. In this work, that necessary condition is shown to be sufficient. As a consequence, we provide a spectral characterization of nonsingular, h-cyclic matrices.

Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

Multi-dimensional Jordan chain and Navier-Stokes equation

Multi-dimensional Jordan chain is presented. It is shown that it admits the finite-component reductions to the Navier-Stokes equation and other important equations of continuous media theory

Preserving spectral properties of structured matrices under structured perturbations

This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues while keeping the Jordan chains unchanged. Moreover, a no spillover structured perturbation of a structured matrix is obtained whose rank is equal to the number of eigenvalues (including multiplicities) which are modified, while preserving the rest of the eigenvalues and the corresponding Jordan chains which need not be known. The specific structured matrices considered in this paper form the Lie algebra and Jordan algebra corresponding to an orthosymmetric scalar product.

Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps

Let \Omega \subset \mathbb{R}^d be a bounded open set with Lipschitz boundary \Gamma . It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in L_2(\Omega) can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from H^{1/2}(\Gamma) into H^{-1/2}(\Gamma) . This result extends the Birman–Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.

Rigorous justification of the Whitham modulation theory for equations of NLS type

AbstractWe study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from Stability of Travelling wave solutions of Nonlinear Dispersive equations of NLS type, where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leismen et al.​to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.

The generalized Birman–Schwinger principle

We prove a generalized Birman–Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a Schrödinger operator) and the associated Birman–Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators. In the course of our analysis we also study algebraic and geometric multiplicities of zeros of strongly analytic operator-valued functions and the associated Jordan chains of generalized eigenvectors. We also relate algebraic multiplicities to the notion of the index of analytic operator-valued functions and derive a general Weinstein–Aronszajn formula for a pair of non-self-adjoint operators.

Finite Rank Perturbations of Linear Relations and Matrix Pencils

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.

Разрешение алгебро-дифференциального уравнения второго порядка относительно производной

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory

The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

Resonance in rarefaction and shock curves: Local analysis and numerics of the continuation method

We describe certain crucial steps in the development of an algorithm for finding the Riemann solution to systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity due to Lax. First, we present a procedure for continuing wave curves beyond points where characteristic speeds coincide, i.e. at wave curve points of maximal co-dimensionality. This procedure requires strict hyperbolicity on both sides of the coincidence locus. Loss of strict hyperbolicity is regularized by means of a Generalized Jordan Chain, which serves to construct a four-fold sub-manifold structure on which wave curves can be continued. Second, we analyze the loss of genuine nonlinearity. We prove a new result: the existence of composite wave curves when the composite wave traverses either the inflection locus or an anomalous part of the non-local composite wave curve. In this sense, we find conditions under which the composite field is well defined and its singularities can be removed, allowing use of our continuation method. Finally, we present numerical examples for a non-strictly hyperbolic system of conservation laws.

Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity

We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components—a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite-length generalized Jordan chains of vectors and that the domain space is the direct sum of these two subspaces. We show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space. If the domain space has a Schauder basis we show that the separated systems of fundamental equations are reduced to two semi-infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the Laurent series coefficients for the resolvent operator on the given annular region.

Nearest Matrix Polynomials With a Specified Elementary Divisor

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 10 September 2019Accepted: 29 June 2020Published online: 01 October 2020Keywordsmatrix polynomial, elementary divisor, Jordan chain, Toeplitz matrixAMS Subject Headings15A18, 65F35, 65F15, 47A56, 15B05, 47J10Publication DataISSN (print): 0895-4798ISSN (online): 1095-7162Publisher: Society for Industrial and Applied MathematicsCODEN: sjmael

Almost Complete Transmission of Low Frequency Waves in a Locally Damaged Elastic Waveguide

It is established that low frequency waves in a symmetric orthotropic elastic waveguide pass almost without distortion through a local perturbation of the straight cylinder, i.e., the reflection coefficient is small and the transmission coefficient slightly differs from 1. The results are obtained by the asymptotic analysis of low frequency waves and the corresponding scattering matrix. The formal asymptotic analysis is also performed for an anisotropic elastic waveguide with asymmetric cross-section, but since the canonical system of Jordan chains for the corresponding operator pencil is too complicated, the known results of the theory of perturbations of nonselfadjoint operators provide asymptotic formulas only under additional elastic and geometric symmetry conditions. We define the polarization matrix and prove its main properties.

On the Kronecker canonical form of a full row rank matrix pencil and applications

Abstract The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

Some results on eigenvalues of finite type, resolvents and Riesz projections

The present paper considers a separable Hilbert space H and a bounded linear operator A:H↦H that has an eigenvalue of finite type at λ0∈C. Using the image and the kernel of certain operators that are defined recursively starting from λ0−A, a construction of the local Smith form and extended canonical system of root functions of λ−A at λ0 is provided. This allows to express the partial multiplicities, the algebraic multiplicity, the Jordan structure and Jordan chains of A at λ0 in terms of the quantities defined by the recursion. Similarly, formulas for the rank, image and kernel of the operators in the principal part of the resolvent (λ−A)−1 at λ0, and thus also of the Riesz projection, are given in terms of the quantities defined by the recursion.

Entanglement classification via integer partitions

In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs.