Operator Pencils Research Articles

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.

(N, ε)-pseudospectra of bounded linear operators on ultrametric Banach spaces

In this paper, we prove that the essential pseudospectrum of bounded linear operator pencils is invariant under perturbation of completely continuous linear operators on ultrametric Banach spaces over a spherically complete field K and we establish a characterization of the essential pseudospectrum of a bounded linear operator pencils by means of the spectra of all perturbed completely continuous operators. Furthermore, we introduce and study the notion of (n,ε)-pseudospectrum of bounded linear operators and the concept of (n,ε)-pseudospectrum of bounded linear operator pencils on ultrametric Banach spaces. We establish some results about them. Finally, several examples are provided.

A multiparametric family of solutions to the Volterra linear integral equation of the first kind

We study the Volterra integral equation of the first kind with an integral operator of order n, a singularity and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms on the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term, we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows one to construct a particular and general solutions to the integro-differential equation in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multi-parameter family of solutions is being constructed for the original integral equation.

Bifurcation and asymptotics of cubically nonlinear transverse magnetic surface plasmon polaritons

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the H1-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

ON THE SPECTRAL THEORY OF A FOURTH-ORDER DIFFERENTIAL PENCIL ON THE WHOLE REAL AXIS

A pencil of fourth-order differential equations on the entire axis with multiple characteristics is considered. Using special solutions of a fourth-order differential equation, the spectrum of a differential pencil is studied. Necessary and sufficient conditions are found for a non-real number to be an eigenvalue of this pencil. It is proved that the operator pencil has no eigenvalues on the real axis.

INVERSE NODAL PROBLEMS FOR PENCILS OF SINGULAR STURM-LIOUVILLE OPERATORS

In this study, some properties of the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point

ON THE MINIMALITY OF A PART OF EIGEN- AND ASSOCIATED VECTORS FOR A CLASS OF THIRD-ORDER QUASI-ELLIPTIC OPE

In the paper, we derive sufficient conditions on the coefficients of a thirdorder quasi-elliptic operator pencil, ensuring the minimality of its system of eigen- and associated vectors corresponding to eigenvalues from the left half-plane. Additionally, we prove a theorem on the minimality of decreasing elementary solutions to a homogeneous equation in a Sobolev-type space.

SPECTRAL PROPERTIES AND INVERSE NODAL PROBLEMS FOR SINGULAR DIFFUSION EQUATION

In this study, some properties for the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues were learned, then the solutions of the inverse problem were given to determine the potential function and parameters of the boundary condition with the help of a dense set of nodal points and lastly we obtain a constructive solution to the inverse problems of this class.

The pseudospectra of linear operator pencils in a Hilbert space

The pseudospectra of linear operator pencils in a Hilbert space

B-essential and B-Weyl spectra of sum of two commuting bounded operators by means operator pencils

B-essential and B-Weyl spectra of sum of two commuting bounded operators by means operator pencils

On the Exponential Stability of the Implicit Differential Systems in Hilbert Spaces

The aim of this research is to study the exponential stability of the stationary implicit system: Ax’(t) + Bx(t) = 0, where A and B are bounded operators in Hilbert spaces. The achieved results are the generalization of Liapounov Theorem for the spectrum of the operator pencil λA + B. We also establish the exponential stability conditions for the corresponding perturbed and quasi-linear implicit systems.

Applications of Nijenhuis Geometry III: Frobenius Pencils and Compatible Non-homogeneous Poisson Structures

We consider multicomponent local Poisson structures of the form {mathcal {P}}_3 + {mathcal {P}}_1, under the assumption that the third-order term {mathcal {P}}_3 is Darboux–Poisson and nondegenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular,we show that each Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges and vertices labelled by numerical marks. They are also naturally related to certain pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the corresponding Nijenhuis pencils.

Fredholm inversion around a singularity: Application to autoregressive time series in Banach space

<abstract><p>This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.</p></abstract>

Solution of a Semi-Boundary-Value Problem for a First-Order Degenerate Partial Differential Equation

A first-order partial differential equation with constant irreversible coefficients in a Banach space is considered. In the particular case of a finite-dimensional space, the initial-boundary-value problem with irreversible matrix coefficients has no solution; hence, we pose Showalter-type conditions. Due to the regularity of the operator pencil, the equation splits into differential equations in subspaces and given conditions lead to initial conditions in subspaces. A solution to the problem is constructed and an example is provided.

To the Problem of Small Oscillations of a System of Two Viscoelastic Fluids Filling Immovable Vessel: Model Problem

In this paper, we study the scalar conjugation problem, which models the problem of small oscillations of two viscoelastic fluids filling a fixed vessel. An initial-boundary value problem is investigated and a theorem on its unique solvability on the positive semiaxis is proven with semigroup theory methods. The spectral problem that arises in this case for normal oscillations of the system is studied by the methods of the spectral theory of operator functions (operator pencils). The resulting operator pencil generalizes both the well-known Kreyn’s operator pencil (oscillations of a viscous fluid in an open vessel) and the pencil arising in the problem of small motions of a viscoelastic fluid in a partially filled vessel. An example of a two-dimensional problem allowing separation of variables is considered, all points of the essential spectrum and branches of eigenvalues are found. Based on this two-dimensional problem, a hypothesis on the structure of the essential spectrum in the scalar conjugation problem is formulated and a theorem on the multiple basis property of the system of root elements of the main operator pencil is proved.

On Spectral and Evolutional Problems Generated by a Sesquilinear Form

On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by a sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations where one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allows us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on a given time interval are obtained. Theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved.

Some spectral sets of linear operators pencils on non-archimedean Banach spaces

In this paper, we define the notions of trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Many results are proved about trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Examples are given to support our work.

On the condition spectrum of linear operator pencils

On the condition spectrum of linear operator pencils

NLS-type equations from quadratic pencil of Lax operators: Negative flows

We formulate and study an integrable model of Nonlinear Schrödinger (NLS)-type through its Lax representation, where one of the Lax operators is quadratic and the other has a rational dependence on the spectral parameter. We discuss the associated spectral problem, the Riemann-Hilbert problem formulation, the conserved quantities, as well as a generalisation for symmetric spaces. In addition we explore the possibilities for modelling with higher order NLS (HNLS) integrable equations and in particular, the relevance of the proposed system.

Problems on Small Motions of Systems of Two Viscoelastic Fluids in Fixed Vessels

In this paper, we study the problem on small motions of two Oldroyd viscoelastic incompressible fluids contained in a fixed vessel. By means of the operator approach, we reduce the original initial-boundary problem to the Cauchy problem for an operator-differential equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary time interval. We deduce the equation for normal oscillations of the considered hydraulic system (the Krein generalized operator pencil).