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Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.


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Journal ArticleDOI
TL;DR: In this paper, the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made is analyzed, and the phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the length of the window passes over critical values is considered.
Abstract: In this paper one analyses the discrete spectrum of an asymmetric pair of two-dimensional quantum waveguides with common boundary in which a window of finite size is made. The phenomenon of new eigenvalues arising at the boundary of the essential spectrum as the length of the window passes over critical values is considered. For the newly arising eigenvalues one constructs asymptotic expansions with respect to the small parameter equal to the difference between the window length and the closest critical value. The behaviour of the spectrum under an unrestricted growth of the length of the window is also under investigation; asymptotic expansions for eigenvalues with respect to the large parameter, the length of the window, are constructed.

55 citations

Journal ArticleDOI
TL;DR: This work examines and develops techniques for obtaining a few selected eigenvalues of the generalized eigenvalue problem Ax = [lambda]Bx, where A and B are n[times]n, nonsymmetric, banded complex matrices, and outlines a procedure to separate the converged eigen values from spurious approximations.

53 citations

Journal ArticleDOI
TL;DR: In this paper, a generalised Rayleigh functional is used that assigns to a vector x a zero of the function T(λ)x, x), where it is assumed that there exists at most one zero.
Abstract: Variational principles for eigenvalues of certain functions whose values are possibly unbounded self-adjoint operators T(λ) are proved. A generalised Rayleigh functional is used that assigns to a vector x a zero of the function T(λ)x, x), where it is assumed that there exists at most one zero. Since there need not exist a zero for all x, an index shift may occur. Using this variational principle, eigenvalues of linear and quadratic polynomials and eigenvalues of block operator matrices in a gap of the essential spectrum are characterised. Moreover, applications are given to an elliptic eigenvalue problem with degenerate weight, Dirac operators, strings in a medium with a viscous friction, and a Sturm-Liouville problem that is rational in the eigenvalue parameter.

51 citations

Journal ArticleDOI
TL;DR: This paper computed the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation, and tried to use this information to predict the performance of iterative methods.
Abstract: The volume integral equation of electromagnetic scattering can be used to compute the scattering by inhomogeneous or anisotropic scatterers. In this paper we compute the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigenvalues lying below the line. We show how the spectrum of the integral operator can be related to the well-posedness of a modified scattering problem. The eigenvalues lying below the line segment arise from resonances in the analytical series solution of scattering by a sphere. The eigenvalues on the line are due to the branch cut of the square root in the definition of the refractive index. We try to use this information to predict the performance of iterative methods. For a normal matrix the initial guess and the eigenvalues of the coefficient matrix determine the rate of convergence of iterative solvers. We show that when the scatterer is a small sphere, the convergence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.

51 citations

Journal ArticleDOI
TL;DR: In this article, the eigenvalues of Schrodinger operators with complex potentials in odd space dimensions were studied and bounds on the total number of eigen values in the case where V decays exponentially at infinity.
Abstract: We study the eigenvalues of Schrodinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity.

50 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
20229
20202
20193
20187
201731