Topic
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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TL;DR: In this article, conditions are given for the spectrum in an eigenvalue problem of the form to be discrete, where and are operators that are odd-homogeneous of degree, acting from a reflexive Banach space into the dual.
Abstract: In this paper conditions are given for the spectrum in an eigenvalue problem of the form to be discrete, where and are operators that are odd-homogeneous of degree (), acting from a reflexive Banach space into the dual. It is proved that the eigenvalues vary monotonically as and vary in the normed linear space of homogeneous operators of degree . Explicit formulas for the eigenvalues and functions are obtained for the case where and are the gradients of the norms in the spaces and ( is a parallelepiped in ). Using these formulas the author obtains estimates for the eigenvalues in homogeneous and asymptotically homogeneous problems with variable coefficients in the space , where is an arbitrary bounded domain in .Bibliography: 12 titles.
12 Sep 2000
TL;DR: In this paper, the existence of eigenoscillations near the system mentioned in the title system is proven and an infinite matrix equation for the coefficients of corresponding expansion is obtained numerically.
Abstract: The existence of eigenoscillations near the system mentioned in the title system is proven. The number of oscillation modes is determined. A classification by groups of possible symmetry is carry out. An infinite matrix equation for the coefficients of corresponding expansion is obtained. This equation is investigated numerically. The plots of eigenvalues versus the length of the cross are obtained. An approximate formula for the eigenvalues is found and investigated. The theory of the self-adjoint operators, the "Dirichlet-Neumenn bracket" and variational methods are used.
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TL;DR: In this paper, the authors proved conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite, based on estimates of the Hilbert-Schmidt norm of semigroup differences.
Abstract: We prove conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite. We use a method for bounding eigenvalues based on estimates of the Hilbert-Schmidt norm of semigroup differences and on complex analysis