Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty $ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.     

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 the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.     

Efficient and stable perfectly matched layer for CEM

An efficient unsplit perfectly matched layer for numerical simulation of electromagnetic waves in unbounded domains is derived via a complex change of variables. In order to surround a Cartesian grid with the PML, the time-dependent PML requires only one (scalar) auxiliary variable in two space dimensions and six (scalar) auxiliary variables in three space dimensions. It is therefore cheap and straightforward to implement. We use Fourier and energy methods to prove the stability of the PML. We extend the stability result to a semi-discrete PML approximated by central finite differences of arbitrary order of accuracy and to a fully discrete problem for the ‘Leap-Frog’ schemes. This makes precise the usefulness of the derived PML model for longtime simulations. Numerical experiments are presented, illustrating the accuracy and stability of the PML.     

Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1–3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.     

An overfilled cavity problem for Maxwell's equations

This paper is concerned with the mathematical analysis of the scattering of a time‐harmonic electromagnetic plane wave by an open and overfilled cavity that is embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. Above the flat ground surface and the open aperture of the cavity, the space is assumed to be filled with a homogeneous medium with a constant permittivity and permeability, whereas the interior of the cavity is filled with some inhomogeneous medium with a variable permittivity and permeability. The scattering problem is modeled as a boundary value problem over a bounded domain, with transparent boundary condition proposed on the hemisphere enclosing the inhomogeneity represented by the cavity. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded electromagnetic cavity scattering problem. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the original scattering problem and that of the truncated PML problem. The error estimate implies that the PML solution converges exponentially to the original cavity scattering problem by increasing either the PML medium parameter or the PML layer thickness. The convergence result is expected to be useful for determining the PML medium parameter in the computational electromagnetic scattering problem. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Navier–Stokes equations with free convection in three‐dimensional unbounded triangular channels

The quaternionic calculus is a powerful tool for treating the Navier–Stokes equations very elegantly and in a compact form, through the evaluation of two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. In this paper, we use special variants of quaternionic‐holomorphic multiperiodic functions in order to obtain explicit formulas for unbounded three‐dimensional parallel plate channels, rectangular block domains and regular triangular channels. Copyright © 2007 John Wiley & Sons, Ltd.     

Essential,Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.     

An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems

We develop an anisotropic perfectly matched layer (PML) method for solving the time harmonic electromagnetic scattering problems in which the PML coordinate stretching is performed only in one direction outside a cuboid domain. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the proposed adaptive anisotropic PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.     

A finite element method for approximating electromagnetic scattering from a conducting object

We provide an error analysis of a fully discrete finite element – Fourier series method for approximating Maxwell's equations. The problem is to approximate the electromagnetic field scattered by a bounded, inhomogeneous and anisotropic body. The method is to truncate the domain of the calculation using a series solution of the field away from this domain. We first prove a decomposition for the Poincaré-Steklov operator on this boundary into an isomorphism and a compact perturbation. This is proved using a novel argument in which the scattering problem is viewed as a perturbation of the free space problem. Using this decomposition, and edge elements to discretize the interior problem, we prove an optimal error estimate for the overall problem.     

Discrete singular convolution method with perfectly matched absorbing layers for the wave scattering by periodic structures

A new computational algorithm is introduced for solving scattering problem in periodic structure. The PML technique is used to deal with the difficulty on truncating the unbounded domain while the DSC algorithm is utilized for the spatial discretization. The present study reveals that the method is efficient for solving the problem.     

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.     

A Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain Part II: Extensions

In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz problems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.     

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius . We also show exponential (in the parameter ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.

Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

     

Examples of embedded eigenvalues for the Dirichlet-Laplacian in domains with infinite boundaries

For certain unbounded domains the Laplace operator with Dirichlet condition is shown to have an unbounded sequence of eigenvalues which are embedded into the essential spectrum. A typical example of such a domain is a locally perturbed cylinder with circular cross-section whose diameter in some bounded subset is greater than at infinity.     

On the self-adjointness of certain compact operators

It is shown that if the singular values of a compact operator T are dominated by those of the real part of T, then T must be self-adjoint. This improves an earlier result of Fong and Tsui. Some remarks on singular value inequalities associated with the Cartesian decomposition of a compact operator are also given.     

Perturbation of semi-Browder operators and stability of Browder's essential defect and approximate point spectrum

In the present paper we investigate the stability of closed densely defined semi-Browder operators under operator perturbations that belong to a perturbation class related to compact operators. Furthermore, we apply the obtained results to give a characterization and to study the stability of Browder's essential approximate point spectrum and Browder's essential defect spectrum.     

On the self-adjointness of certain compact operators

It is shown that if the singular values of a compact operator T are dominated by those of the real part of T, then T must be self-adjoint. This improves an earlier result of Fong and Tsui. Some remarks on singular value inequalities associated with the Cartesian decomposition of a compact operator are also given.     

Über das wesentliche Spektrum zum Eigenwertproblem der automorphen Formen

The eigenvalue problem of automorphic forms is the problem to determine the spectral behaviour of a certain second order elliptic partial differential operator having its domain of definition in an appropriate Hilbert space of square integrable functions defined in the upper halfplane. This Hilbert space depends on a discontinuous group G, a real parameter k, and a multiplier system v on G of weight 2k. We prove: The essential spectrum of the self-adjoint linear operator under consideration contains the interval \([\frac{1}{4}, \infty \rangle\) if G has a singular cusp or if G is a Fuchsian group of the second kind. If G has a sufficiently large fundamental domain, the essential spectrum of our differential operator is non-empty and unbounded above.