Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

Riesz basis property and exponential stability for one-dimensional thermoelastic system with variable coefficients

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case} We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a result, the exponential stability of the system is concluded.

Stability of some essential B-spectra of pencil operators and application

In this paper, we give some results on the essential B-spectra of a linear operator pencil, which are used to determine the essential B-spectra of an integro-differential operator with abstract boundary conditions in the Banach space Lp([−a, a] × [−1, 1]), p ≥ 1 and a > 0.

Solvability conditions in the analytical form of a descriptor system of partial differential equations

A system of first-order partial differential-algebraic equations in a Banach space with constant degenerate operators in the case of a regular operator pencil is considered. In this case, under some additional condition, the original system splits into two subsystems in disjoint subspaces in order to search for the projections of the original unknown function in the subspaces. The matching conditions for the parameters of the systems are identified. A solution of the considered system of differential-algebraic equations is constructed.

To the Problem on 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 S. G. Kreyns 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.

The condition pseudospectrum of a operator pencil

In this paper, we establish some properties concerning the condition pseudospectrum of the linear operator pencils [Formula: see text] when, [Formula: see text] is not necessarily invertible. Also, we give some results related to the generalized condition pseudospectra and the generalized essential condition pseudospectra of linear operators. We start by studying the stability of these condition pseudospectra and some characterization.

Spectra of operator pencils with small 𝒫𝒯-symmetric periodic perturbation

We study the spectrum of a quadratic operator pencil with a small 𝒫𝒯-symmetric periodic potential and a fixed localized potential. We show that the continuous spectrum has a band structure with bands on the imaginary axis separated by usual gaps, while on the real axis, there are no gaps but at certain points, the bands bifurcate into small parabolas in the complex plane. We study the isolated eigenvalues converging to the continuous spectrum. We show that they can emerge only in the aforementioned gaps or in the vicinities of the small parabolas, at most two isolated eigenvalues in each case. We establish sufficient conditions for the existence and absence of such eigenvalues. In the case of the existence, we prove that these eigenvalues depend analytically on a small parameter and we find the leading terms of their Taylor expansions. It is shown that the mechanism of the eigenvalue emergence is different from that for small localized perturbations studied in many previous works.