共查询到20条相似文献,搜索用时 15 毫秒
1.
S. A. Nazarov 《Computational Mathematics and Mathematical Physics》2012,52(3):448-464
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. 相似文献
2.
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. 相似文献
3.
S. A. Nazarov 《Functional Analysis and Its Applications》2013,47(3):195-209
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. 相似文献
4.
S. A. Nazarov 《Computational Mathematics and Mathematical Physics》2013,53(6):702-720
It is assumed that a trapped mode (i.e., a function decaying at infinity that leaves small discrepancies of order ? ? 1 in the Helmholtz equation and the Neumann boundary condition) at some frequency κ0 is found approximately in an acoustic waveguide Ω0. Under certain constraints, it is shows that there exists a regularly perturbed waveguide Ω? with the eigenfrequency κ? = κ0 + O(?). The corresponding eigenvalue λ? of the operator belongs to the continuous spectrum and, being naturally unstable, requires “fine tuning” of the parameters of the small perturbation of the waveguide wall. The analysis is based on the concepts of the augmented scattering matrix and the enforced stability of eigenvalues in the continuous spectrum. 相似文献
5.
S. A. Nazarov 《Theoretical and Mathematical Physics》2011,167(2):606-627
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. 相似文献
6.
《偏微分方程通讯》2013,38(4):539-565
Abstract The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right. 相似文献
7.
S. A. Nazarov 《Computational Mathematics and Mathematical Physics》2009,49(2):323-333
A three-dimensional periodic elastic waveguide is constructed whose continuous spectrum (the frequencies that admit propagating waves) contains a gap, i.e., an interval that has its ends in the continuous spectrum but contains at most a discrete spectrum. The waveguide consists of an infinite chain of massive bodies connected by short thin links, and its surface is assumed to be free. The method for detecting a gap also applies to plane problems, including scalar ones. Periodic elastic waveguides with different shapes or contrasting properties are indicated in which a gap can also be detected. 相似文献
8.
F. Peherstorfer 《Journal of Functional Analysis》2009,256(7):2157-2210
We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: (1) the class of twosided CMV matrices acting in l2, whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in l2) behaves asymptotically as the basis associated with the free system. (2) From the point of view of the scattering problem, the most natural class of CMV matrices is that one in which (a) the scattering data determine the matrix uniquely and (b) the associated Gelfand-Levitan-Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an A2 kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar conditions close to the optimal ones are given directly in terms of the scattering data. 相似文献
9.
A. A. Arsen'ev 《Theoretical and Mathematical Physics》2003,136(3):1336-1345
We study the asymptotic behavior of the residue at the pole of the analytic continuation of the scattering matrix as the imaginary part of the pole tends to zero in the case where the phase space of a quantum mechanical system is a direct sum of two spaces and the nonperturbed evolution operator reduces each of these spaces and has a discrete spectrum in one of them and a continuous spectrum in the other. The perturbation operator mixes the subspaces and generates a resonance. We prove that under certain symmetry conditions in such a system, the scattering amplitude changes sharply in a neighborhood of the real part of the pole of the scattering matrix, and the system demonstrates tunneling or a resonance of the scattering amplitude. 相似文献
10.
V. I. Derguzov 《Journal of Mathematical Sciences》1995,77(4):3295-3302
A waveguide operator is defined. It is proved that its spectrum coincides with the spectrum of a lightguide. The classification
of singular points of the continuous spectrum is given. Invariant subspaces of the waveguide operator are distinguished that
are related to an interval of the continuous spectrum without singular points. Bibliography: 9 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 51–62. 相似文献
11.
In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ?X. In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry. 相似文献
12.
Yoichi Uetake 《Integral Equations and Operator Theory》2008,60(2):271-288
We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full
modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all
functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some
properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous
spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the
the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami
operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of
the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.
相似文献
13.
We outline a method to compute the solution in the frequency–domain for scattering in a waveguide by exploiting symmetry. The method is illustrated by considering a simple scattering example, where soft hard boundary conditions are alternated. We show how the straightforward mode matching or eigenfunction matching solution can be easily converted to scattering and transmission matrices when symmetry is exploited. We then show how the solution for two scatterers can be found explicitly, using symmetry which allows validation of our subsequent solution by scattering matrices. We also give a series of identities which the scattering matrix must satisfy for further numerical validation. Using these frequency–domain solutions we compute the time-domain scattering by incident Gaussian wave–packets. 相似文献
14.
M. D. Malykh 《Computational Mathematics and Modeling》2003,14(4):379-382
The existence of waveguide trapped modes is considered. We prove that trapped modes embedded in a continuous spectrum are unstable in the presence of small real perturbations of the waveguide filling. 相似文献
15.
E. M. Il'in 《Journal of Mathematical Sciences》1987,36(3):417-419
In the case of the scattering problem on a wedge and on a screen, for a certain class of boundary conditions, one constructs explicitly the wave operators and one establishes their completeness. It is shown that a modified scattering matrix (including additionally the reflection operator) is a unitary operator with a pure point spectrum. In the case of a screen, the standard S-matrix is unitary. For Dirichlet (Neumann) boundary conditions, the S-matrix is reduced explicitly to a diagonal form. The spectrum of the S-matrix is simple, absolutely continuous, filling the lower (upper) semicircumference.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 193–197, 1982. 相似文献
16.
S. A. Nazarov 《Mathematical Notes》2010,87(5-6):738-756
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. 相似文献
17.
** Email: darkovolkov{at}yahoo.ca, darko{at}wpi.edu
The scattering matrix for a perfectly conducting electricalcylinder (or a sound hard obstacle) in a waveguide is unitary.This is a well-known result which is a consequence of the conservationof power. When a numerical method is employed to approximatethe reflection and transmission coefficients of the cylinder,an approximate scattering matrix can be constructed. An integralequation of the second kind for an unknown density can be solved,and the density can then be used for computing the entries ofthe approximate scattering matrix. We show that this approximatematrix is unitary for cylinders of symmetric cross-section,regardless of the order of the approximation. In the nonsymmetriccase, the approximate scattering matrix still satisfies a conservationof energy condition, albeit in an unfamiliar form. As the orderof the approximation is increased, conservation of energy isalso satisfied in the more familiar form to machine accuracy. 相似文献
18.
K. V. Malykhin 《Journal of Mathematical Sciences》1989,45(3):1230-1235
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. 相似文献
19.
V. I. Derguzov 《Journal of Mathematical Sciences》1989,45(3):1199-1205
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. 相似文献
20.
V. I. Derguzov 《Journal of Mathematical Sciences》1986,35(1):2170-2179
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. 相似文献