Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space

In this paper, we develop a perturbation analysis for stability spectra (Lyapunov exponents and Sacker–Sell spectrum) for products of operators on a Hilbert space (both real and complex) based upon the discrete QR technique. Error bounds are obtained in both the integrally separated and non-integrally separated cases that correspond to distinct and multiple eigenvalues, respectively, for a single linear operator. We illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation (the time varying analogue of gaps between eigenvalues) determines the sensitivity of the stability spectra to perturbation.     

Discrete spectrum in the gaps for perturbations of periodic Jacobi matrices

In the paper we study the problem of the finiteness of the discrete spectrum for operators generated by perturbations of periodic Jacobi matrices. In particular, estimate formulae for the number of the eigenvalues created by perturbations in the gaps of the unperturbed operator are established.     

Algorithm for min-range multiplication of affine forms

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.     

Perturbation of a periodic operator by a narrow potential

We consider perturbations of a second-order periodic operator on the line; the Schr?dinger operator with a periodic potential is a specific case of such an operator. The perturbation is realized by a potential depending on two small parameters, one of which describes the length of the potential support, and the inverse value of other corresponds to the value of the potential. We obtain sufficient conditions for the perturbing potential to have eigenvalues in the gaps of the continuous spectrum. We also construct their asymptotic expansions and present sufficient conditions for the eigenvalues of the perturbing potential to be absent.     

Remarks on Non-Linear Spectral Perturbation

We are interested in some aspects of the perturbation effects in the spectrum of a real nonnormal matrix A under linear perturbations. We discuss some known results and we use them to justify some recent experimental observations. Moreover, we demonstrate that the qualitative behavior of the eigenvalues of A under linear perturbations may be predicted by inspecting the spectral radius of a related matrix. Then, we show how this information can be used to analyze the quality of the approximation of a projection method and to justify the presence of unexpected approximate eigenvalues.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Fredholm theory for pairs of closed subspaces of a Banach space

We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces.     

On the Spectral Stability of Kinks in 2D Klein–Gordon Model with Parity‐Time‐Symmetric Perturbation

In a series of recent works by Demirkaya et al., stability analysis for the static kink solutions to the one‐dimensional continuous and discrete Klein–Gordon equations with a ‐symmetric perturbation has been performed. In the present paper, we study two‐dimensional (2D) quadratic operator pencil with a small localized perturbation. Such an operator pencil is motivated by the stability problem for the static kink in 2D Klein–Gordon field taking into account spatially localized ‐symmetric perturbation, which is in the form of viscous friction. Viscous regions with positive and negative viscosity coefficient are balanced. For the considered operator pencil, we show that its essential spectrum has certain critical points generating eigenvalues under the perturbation. Our main results are sufficient conditions ensuring the existence or absence of such eigenvalues as well as the asymptotic expansions for these eigenvalues if they exist.     

Error estimate of eigenvalues of discrete linear Hamiltonian systems with small perturbation

In this paper, eigenvalues of perturbed discrete linear Hamiltonian systems are considered. A new variational formula of eigenvalues is first established. Based on it, error estimates of eigenvalues of systems with small perturbation are given under certain non-singularity conditions. Small perturbations of the coefficient functions, the weight function and the coefficients of the boundary condition are all involved. As a direct consequence, continuous dependence of eigenvalues on boundary value problems is obtained under the non-singularity conditions. In addition, two examples are presented to illustrate the necessity of the non-singularity conditions and the complexity of the problem in the singularity case.     

Transmission eigenvalue problem for inhomogeneous absorbing media with mixed boundary condition

Consider the transmission eigenvalue problem for the wave scattering by a dielectric inhomogeneous absorbing obstacle lying on a perfect conducting surface. After excluding the purely imaginary transmission eigenvalues, we prove that the transmission eigenvalues exist and form a discrete set for inhomogeneous non-absorbing media, by using analytic Fredholm theory. Moreover, we derive the Faber-Krahn type inequalities revealing the lower bounds on real transmission eigenvalues in terms of the media parameters. Then, for inhomogeneous media with small absorption, we prove that the transmission eigenvalues also exist and form a discrete set by using perturbation theory. Finally, for homogeneous media, we present possible components of the eigenvalue-free zone quantitatively, giving the geometric understanding on this problem.     

Weyl's theorem for closed operators on Fréchet spaces

A celebrated theorem of H. Weyl asserts that if A is a normal operator on a Hilbert space X, then the points in the spectrum of A which can be removed by perturbing A with a compact operator are precisely the eigenvalues of finite multiplicity which are isolated points of the spectrum of A. In these notes we develop appropriate Fredholm theory that enables us to provide interesting sufficiency and necessary conditions for a closed operator on a Fréchet space to satisfy Weyl's theorem. We complement and extend results of L. A. Coburn, V. Istr??escu, S. K. Berberian, and M. Schechter, respectively.     

Relative essential spectra involving relative demicompact unbounded linear operators

In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter's and approximate essential spectrum.     

Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.     

Coperturbation function and lower semi-Browder multivalued linear operators

In this article, we investigate the perturbation theory of lower semi-Browder and Browder linear relations. Our approach is based on the concept of a coperturbation function for linear relations in order to establish some perturbation theorems and deduce the stability under strictly cosingular operator perturbations. Furthermore, we apply the obtained results to study the invariance and the characterization of Browder's essential defect spectrum and Browder's essential spectrum.     

Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrödinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrödinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.     

Eigenvalues of an elliptic system

We describe the spectrum of a non-self-adjoint elliptic system on a finite interval. Under certain conditions we find that the eigenvalues form a discrete set and converge asymptotically at infinity to one of several straight lines. The eigenfunctions need not generate a basis of the relevant Hilbert space, and the larger eigenvalues are extremely sensitive to small perturbations of the operator. We show that the leading term in the spectral asymptotics is closely related to a certain convex polygon, and that the spectrum does not determine the operator up to similarity. Two elliptic systems which only differ in their boundary conditions may have entirely different spectral asymptotics. While our study makes no claim to generality, the results obtained will have to be incorporated into any future general theory. Received: 15 August 2001 / in final form: 11 February 2002 / Published online: 24 February 2003     

Birkhoff's theorem and multidimensional numerical range

We show that, under certain conditions, Birkhoff's theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections. Using this result, we study the relation between the spectrum of a self-adjoint operator A and its multidimensional numerical range. It turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator A. Every collection of eigenvalues which can be obtained by the Rayleigh-Ritz formula generates an extreme point of the multidimensional numerical range. However, it may also have other extreme points.     

一致Fredholm指标算子及广义(ω′)性质

本文给出了一致Fredholm指标算子的定义及判定,同时定义了Weyl型定理的一种新变化:广义(ω')性质.根据一致Fredholm指标性质定义出一种新的谱集,通过该谱集给出了Hilbert空间上有界线性算子满足广义(ω')性质的充要条件,并且研究了广义(ω')性质的摄动,还研究了算子的亚循环性和广义(ω')性质之间的关系.     

RELATIVE ESSENTIAL SPECTRA INVOLVING RELATIVE DEMICOMPACT UNBOUNDED LINEAR OPERATORS

In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.     

A certain approach to the problem of regularization on the basis of the perturbation of linear operators

We consider a method for constructing the solutions of a linear Fredholm operator equation that is regularized by means of a special perturbation of the equation by a linear operator.Translated from Matematicheskii Zametki, Vol. 20, No. 5, pp. 747–752, November, 1976.The author is grateful to M. A. Krasnosel'skill for discussing the results of this article.