Quasi-periodic Boundary Value Problem Research Articles

The functional analytic approach for quasi-periodic boundary value problems for the Helmholtz equation

The functional analytic approach for quasi-periodic boundary value problems for the Helmholtz equation

Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

The paper is concerned with the completeness property of rank one perturbations of the unperturbed operators generated by special boundary value problems (BVP) for the following $$2 \times 2$$ system 0.1 $$\begin{aligned}&L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 &{}\quad 0 \\ 0 &{}\quad b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{aligned}$$ on a finite interval assuming that the potential matrix Q is summable, and $$b_1 b_2^{-1} \notin \mathbb {R}$$ (essentially non-Dirac type case). We assume that the unperturbed operator generated by a BVP belongs to one of the following three subclasses of the class of spectral operators: (a) normal operators; (b) operators similar either to a normal or almost normal; (c) operators that meet Riesz basis property with parentheses; We show that in each of the three cases there exists (non-unique) operator generated by a quasi-periodic BVP and its certain rank-one perturbations (in the resolvent sense) generated by special BVPs which are complete while their adjoint are not. In connection with the case (b) we investigate Riesz basis property of quasi-periodic BVP under certain assumptions on the potential matrix Q. We also find a simple formula for the rank of the resolvent difference for operators corresponding to two BVPs for $$n \times n$$ system in terms of the coefficients of linear boundary forms.

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.

Asymptotic behaviour of the discrete spectrum of a quasi-periodic boundary value problem for a two-dimensional hyperbolic equation

This paper is concerned with the asymptotic properties of the discrete spectrum of two-dimensional self-adjoint operators of hyperbolic type. For the operator of the model quasi-periodic boundary value problem associated with a self-adjoint hyperbolic equation with smooth coefficients on a two-dimensional torus we obtain an asymptotic formula for the distribution function of the eigenvalues.Bibliography: 9 titles.

A study on quasi-periodic boundary value problems for nonlinear higher order impulsive differential equations

A study concerning the solvability of the quasi-periodic boundary value problems for nonlinear higher order impulsive differential equations is made. Sufficient conditions for the existence of at least one solution to the problems are obtained. Examples are presented to illustrate the main results.

Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain

SynopsisIn this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the gratingwith respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables (x, y, δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

On quasiperiodic boundary value problems and their applications in the theory of elasticity

The fundamental boundary value problems of analytic function theory are considered on certain systems of contours possessing translational symmetry: the Riemann problem on an oblique lattice of arbitrary contours, the Hubert problem and the mixed problem for a half-plane, the Dirichlet problem for a plane with a periodic system of slits on a line. In contrast to /1/, where the listed problems are solved under the condition of periodicity of their coefficients and free terms, this condition is here imposed only on the coefficients. By applying a discrete Fourier transform and periodicity of the boundary conditions for an elementary cell, the formulated quasiperiodic problems are reduced to periodic problems and are solved in closed form. The results obtained are used to solve (in quadratures) new mixed problems of elasticity theory in translationally symmetric domains with nonperiodic boundary conditions.

A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent

SynopsisA quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.