Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle

Perturbations of an eigenvalue in the continuous spectrum of the Neumann problem for the Laplacian in a strip waveguide with an obstacle symmetric about the midline are studied. Such an eigenvalue is known to be unstable, and an arbitrarily small perturbation can cause it to leave the spectrum to become a complex resonance point. Conditions on the perturbation of the obstacle boundary are found under which the eigenvalue persists in the continuous spectrum. The result is obtained via the asymptotic analysis of an auxiliary object, namely, an augmented scattering matrix.     

Perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle

Doklady Mathematics -     

The resolvent of a periodic optical waveguide the continuous spectrum

We consider a two-dimensional optical waveguide with periodic media interface. We study the resolvent of the waveguide in a neighborhood of a purely imaginary point of the spectral parameter. We prove that the resolvent exists on the subspace of functions orthogonal ina certain sense to the singular functions of the continuous spectrum. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 79–89.     

Opening of a gap in the continuous spectrum of a periodically perturbed waveguide

It is established that a small periodic singular or regular perturbation of the boundary of a cylindrical three-dimensional waveguide can open up a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator in the resulting periodic waveguide. A singular perturbation results in the formation of a periodic family of small cavities while a regular one leads to a gentle periodic bending of the boundary. If the period is short, there is no gap, while if it is long, a gap appears immediately after the first segment of the continuous spectrum. The result is obtained by asymptotic analysis of the eigenvalues of an auxiliary problem on the perturbed cell of periodicity.     

Eigenfunctions of the continuous spectrum of a waveguide with periodic boundary

One proves the existence of the eigenfunctions of the continuous spectrum of a two-dimensional waveguide with periodic boundary. One carries out a normalization of the eigenfunctions of the continuous spectrum relative to an indefinite inner product.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 154–160, 1986.     

Singular points of the continuous spectrum of a two-dimensional periodic waveguide

One investigates the eigenfunctions of the continuous spectrum of a waveguide in the neighborhood of a special singular point of the continuous spectrum. One proves the existence of a complete collection of eigenfunctions of the continuous spectrum in the neighborhood of such a point.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 116–123, 1986.     

Eigenfunctions of the continuous spectrum of a two-dimensional periodic optical waveguide

One proves the existence of the eigenfunctions of the continuous spectrum of a two-dimensional periodic optical waveguide. One gives a normalization of the eigenfunctions of the continuous spectrum relative to an indefinite inner product. One defines the concept of the genus of the multipliers of a Hamiltonian equation, corresponding to the continuous spectrum of the optical waveguide.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 18–34, 1984.     

Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

We establish that by choosing a smooth local perturbation of the boundary of a planar quantum waveguide, we can create an eigenvalue near any given threshold of the continuous spectrum and the corresponding trapped wave exponentially decaying at infinity. Based on an analysis of an auxiliary object, a unitary augmented scattering matrix, we asymptotically interpret Wood’s anomalies, the phenomenon of fast variations in the diffraction pattern due to variations in the near-threshold wave frequency.     

Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace–Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in particular, does not require any global assumptions on the topology or the curvature, unlike previous papers on the same topic.     

On the asymptotics and stability of the point spectrum of a waveguide with thin shielding obstacle

Some explicit conditions are found for the existence and absence of an eigenvalue in the interval (0, π ²) of the continuous spectrum of the Neumann problem for the Laplace operator in the unit strip with a thin (of width O(ε)) symmetric screen, which, as ε → +0, shrinks into a line segment perpendicular to the sides of the strip. An asymptotics of this eigenvalue is constructed, as well as the asymptotics of the reflection coefficient, which describes Wood’s anomalies, namely, quick changes of the diffraction pattern near a frequency threshold in the continuous spectrum. Bibliography: 32 titles.     

Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide

In this paper, we provide uniform bounds for convergence rates of the low frequencies of a parametric family of problems for the Laplace operator posed on a rectangular perforated domain of the plane of height H. The perforations are periodically placed along the ordinate axis at a distance $O(\epsilon )$ between them, where ε is a parameter that converges toward zero. Another parameter η, the Floquet-parameter, ranges in the interval $[-\pi ,\pi ]$. The boundary conditions are quasi-periodicity conditions on the lateral sides of the rectangle and Neumann over the rest. We obtain precise bounds for convergence rates which are uniform on both parameters ε and η and strongly depend on H. As a model problem associated with a waveguide, one of the main difficulties in our analysis comes near the nodes of the limit dispersion curves.     

Asymptotic formula for an eigenvalue of the dirichlet problem in a cranked waveguide

The asymptotic behavior of an eigenvalue of the Dirichlet problem for a spectral Helmholtz equation in a two-dimensional cranked acoustic waveguide with yielding walls or in a quantum waveguide is obtained. A waveguide is thought of as a cranked strip, but the boundary value problem is posed in a straight strip of unit width with wedge-shaped notches, with appropriate conjugation conditions on the edges of the notches, which provide for a smooth wave field after the initial form of the waveguide is restored. The bend angles are assumed to be small; i.e., the wedge-shaped notches are supposed to be thin, the asymptotic behavior is built from the corresponding small geometric parameter.     

Asymptotics of an eigenvalue on the continuous spectrum of two quantum waveguides coupled through narrow windows

Conditions under which two planar identical waveguides coupled through narrow windows of width ? ? 1 have an eigenvalue on the continuous spectrum are obtained. It is established that the eigenvalue appears only for certain values of the distance between the windows: for each sufficiently small ? > 0, there exists a sequence $(2N - 1)/\sqrt 3 + O(\varepsilon )$ of such distances; here N = 1, 2, 3, .... The result is obtained by the asymptotic analysis of an auxiliary object, namely, the augmented scattering matrix.     

Criteria for stability of the point spectrum under completely continuous perturbations

We show that a number is an eigenvalue of the operator T+C for an arbitrary compact perturbation C if and only if the operator T -I is semi-Fredholm and ind (T-I)>0.Translated from Matematicheskie Zametki, Vol. 18, No. 4, pp. 601–607, October, 1975.     

An automorphism with simple,continuous spectrum not having the group property

An example is constructed of an automorphism with simple and continuous spectrum which does not have the group property.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 323–326, March, 1969.     

Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.