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 -     

On a method for computing waveguide scattering matrices in the presence of point spectrum

A waveguide occupies a domain G in ? ⁿ⁺¹, n ? 1, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter µ. As an approximation to a row of the scattering matrix S(µ) we suggest a minimizer of a quadratic functional J ^R (·, µ). To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance R, the waveguide outlets to infinity. It is proved that, if a finite interval [µ₁, µ₂] of the continuous spectrum contains no thresholds, then, as R → ∞, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to µ ∈ [µ₁, µ₂]. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity.     

On the asymptotics of locally dependent point processes

We investigate a family of approximating processes that can capture the asymptotic behaviour of locally dependent point processes. We prove two theorems presented to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approximating processes can circumvent the technical difficulties encountered in compound Poisson process approximation (see Barbour and Månsson (2002) [10]) and our approximation error bound decreases when the mean number of the random events increases, in contrast to the increasing of bounds for compound Poisson process approximation.     

On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide

It is shown in the paper that, under several orthogonality and normalization conditions and a proper choice of accessory parameters, a simple eigenvalue lying between thresholds of the continuous spectrum of the Dirichlet problem in a domain with a cylindrical outlet to infinity is not taken out from the spectrum by a small compact perturbation of the Helmholtz operator. The result is obtained by means of an asymptotic analysis of the augmented scattering matrix.     

On a two-point boundary value problem for the Sturm-Liouville operator with a nonclassical asymptotics of the spectrum

We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L ₂(0, π) such that the multiplicities of the eigenvalues λ _n monotonically tend to infinity as n → ∞.     

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.     

Nonstationary normal waves in a thin,curved waveguide in a multiray zone (uniform asymptotics)

In a thin waveguide with properties that vary along its course, the propagation of nonstationary normal waves in the presence of caustics for space-time rays is considered. The connection of the critical section of the waveguide with such caustics is determined. Uniform asymptotic formulas are obtained for the wave field in a multiray zone, and the passage into geometric rays outside a neighborhood of the caustics is traced.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 127–137, 1980.In conclusion, we note that the matrix machinery used in the present work is not connected with the special nature of the present problem and can always be applied when the initial Ansatz contains a collection of special functions which go over into one another under differentiation.     

Binomial asymptotics of the spectrum of a boundary-value problem

Functional Analysis and Its Applications -     

Unilateral contact of a plate with a thin elastic obstacle

We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.     

On the point spectrum of Dirac operators

We study the decay of eigenfunctions and we give conditions for the absence of eigenvalues embedded in the continuous spectrum for Dirac Hamiltonians with long range and locally singular potential.     

On asymptotics for the spectrum of the product of two random rectangular matrices

On assuming the Lindeberg condition, we prove the convergence of the expected spectral distribution of the product of two independent random rectangular matrices with independent entries to a certain distribution on the unit disk in the complex plane. We obtain an explicit expression for the density of the limit distribution.     

On the spectrum of a weak class of operator pencils of waveguide type

The paper introduces a new class of two parameter non‐overdamped operator pencils arising from evolution equations. We investigate spectral properties, including variational principles for “interior” points of the spectrum. Examples leading to pencils of the new class are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures.     

The asymptotics of the return map of a singular point with fixed Newton diagram

We exhibit a large class of nondegenerate singular points in which necessary and sufficient conditions are given for monodromy. We compute the generalized first Lyapunov value, which is expressed in terms of the Newton diagram of the singular point. The computational algorithm proposed is based on writing the return map as the composition of transition mappings constructed using the diagram. The nonvanishing of the generalized first Lyapunov value is a sufficient condition for the existence of a focus. Bibliography: 8 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 156–177, 1991.