共查询到20条相似文献,搜索用时 15 毫秒
1.
2.
Julia Elyseeva 《Applied Mathematics Letters》2012,25(11):1809-1814
We develop an analog of classical oscillation theory for discrete symplectic eigenvalue problems with general self-adjoint boundary conditions which, rather than measuring of the spectrum of one single problem, measures the difference between the spectra of two different problems. We prove formulas connecting the numbers of eigenvalues in a given interval for two symplectic eigenvalue problems with different self-adjoint boundary conditions. We derive as corollaries generalized interlacing properties of eigenvalues. 相似文献
3.
Paul Binding 《Journal of Differential Equations》1983,49(3):331-343
We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem , 0 ≠ xm?Hm, m = 1 … k. (1) Here Tm and Vmn are self-adjoint operators on Hilbert spaces Hm. The index is based on the number of negative eigenvalues of Tm ? ∑n = 1kλnVmn and on the sign of the determinant δ0 with (m, n)th entry (xm, Vmnxm). We assume that certain cofactors of δ0 are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ0 is positive. 相似文献
4.
The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary conditions, with the set of Cauchy data for the differential operator. We apply this methodology to constrained eigenvalue problems, in which the operator is restricted to a (not necessarily invariant) subspace. The Maslov index is defined and used to compute the Morse index of the constrained operator. We then prove a constrained Morse index theorem, which says that the Morse index of the constrained problem equals the number of constrained conjugate points, counted with multiplicity, and give an application to the nonlinear Schrödinger equation. 相似文献
5.
6.
Yu. V. Eliseeva 《Russian Mathematics (Iz VUZ)》2011,55(11):71-75
In this paper we consider a discrete symplectic eigenvalue problem with separated boundary conditions and obtain formulas
for the number of eigenvalues on a given interval of the variation of the spectral parameter. In addition, we compare the
spectra of two symplectic eigenvalue problems with different separated boundary conditions. 相似文献
7.
Yuji Nakatsukasa 《Linear algebra and its applications》2010,432(1):242-248
Weyl-type eigenvalue perturbation theories are derived for Hermitian definite pencils A-λB, in which B is positive definite. The results provide a one-to-one correspondence between the original and perturbed eigenvalues, and give a uniform perturbation bound. We give both absolute and relative perturbation results, defined in the standard Euclidean metric instead of the chordal metric that is often used. 相似文献
8.
9.
10.
11.
Charles V Coffman 《Journal of Functional Analysis》1973,14(3):237-252
We treat the eigenvalue problem Ax = λBx, where A and B are odd potential operators, A is strictly monotone, bounded, coercive, and continuously invertible, and B is monotone and compact. A naturally defined iteration operator is employed, together with the Lyusternik-Schnirelman theory, to prove the existence of infinitely many nontrivial eigenfunctions. With the possible exception of the multiplicity assertion the results which we obtain are not new. The method which we use, however, has not been applied before to problems of this type. It exploits both the potential character and the monotonicity of the operators and makes the treatment of the infinite dimensional problem essentially as simple as that of its finite dimensional analog. This simplification results primarily from the compactness properties of the iteration operator. 相似文献
12.
Shmuel Friedland 《Linear algebra and its applications》1975,12(2):127-137
Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ1,…,λn} in C. The number of different matrices D is at most n!. 相似文献
13.
Oscillation spectral properties (the number of zeros, their alternation for eigenfunctions, the simplicity of the spectrum,
etc.) are described for the Sturm-Liouville problem with generalized coefficients.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 95–99, January, 2008. 相似文献
14.
D. G. Rakhimov 《Differential Equations》2011,47(9):1372-1377
We consider the computation of simple and multiple eigenvalues and the corresponding eigenelements of a multiparameter eigenvalue problem in the sense of Atkinson by the reduction pseudoperturbation method. Namely, we regularize the original operator functions by finite-dimensional linear operators, thus reducing the case of multiple eigenvalues to that of simple ones. 相似文献
15.
This work deals with various finite algorithms that solve two special Structured Inverse Eigenvalue Problems (SIEP). The first
problem we consider is the Jacobi Inverse Eigenvalue Problem (JIEP): given some constraints on two sets of reals, find a Jacobi
matrix J (real, symmetric, tridiagonal, with positive off-diagonal entries) that admits as spectrum and principal subspectrum
the two given sets. Two classes of finite algorithms are considered. The polynomial algorithm which is based on a special
Euclid–Sturm algorithm (Householder's terminology) and has been rediscovered several times. The matrix algorithm which is
a symmetric Lanczos algorithm with a special initial vector. Some characterization of the matrix ensures the equivalence of
the two algorithms in exact arithmetic. The results of the symmetric situation are extended to the nonsymmetric case. This
is the second SIEP to be considered: the Tridiagonal Inverse Eigenvalue Problem (TIEP). Possible breakdowns may occur in the
polynomial algorithm as it may happen with the nonsymmetric Lanczos algorithm. The connection between the two algorithms exhibits
a similarity transformation from the classical Frobenius companion matrix to the tridiagonal matrix. This result is used to
illustrate the fact that, when computing the eigenvalues of a matrix, the nonsymmetric Lanczos algorithm may lead to a slow
convergence, even for a symmetric matrix, since an outer eigenvalue of the tridiagonal matrix of order n − 1 can be arbitrarily
far from the spectrum of the original matrix.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
16.
Jian Deng 《Journal of Differential Equations》2006,225(1):57-89
An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain. 相似文献
17.
Rolf Rannacher 《Numerische Mathematik》1979,33(1):23-42
The paper deals with nonconforming finite element methods for approximating fourth order eigenvalue problems of type
2
w=w. The methods are handled within an abstract Hilbert space framework which is a special case of the discrete approximation schemes introduced by Stummel and Grigorieff. This leads to qualitative spectral convergence under rather weak conditions guaranteeing the basic properties of consistency and discrete compactness for the nonconforming methods. Further asymptotic error estimates for eigenvalues and eigenfunctions are derived in terms of the given orders of approximability and nonconformity. These results can be applied to various nonconforming finite elements used by Adini, Morley, Zienkiewicz, de Veubeke e.a. This is carried out for the simple elements of Adini and Morley and is illustrated by some numerical results at the end. 相似文献
18.
Luís M. Fernandes Joaquim J. Júdice Hanif D. Sherali Maria A. Forjaz 《Computational Optimization and Applications》2014,59(1-2):113-134
In this paper, we discuss the solution of linear and quadratic eigenvalue complementarity problems (EiCPs) using an enumerative algorithm of the type introduced by Júdice et al. (Optim. Methods Softw. 24:549–586, 2009). Procedures for computing the interval that contains all the eigenvalues of the linear EiCP are first presented. A nonlinear programming (NLP) model for the quadratic EiCP is formulated next, and a necessary and sufficient condition for a stationary point of the NLP to be a solution of the quadratic EiCP is established. An extension of the enumerative algorithm for the quadratic EiCP is also developed, which solves this problem by computing a global minimum for the NLP formulation. Some computational experience is presented to highlight the efficiency and efficacy of the proposed enumerative algorithm for solving linear and quadratic EiCPs. 相似文献
19.
Jing Lin 《Journal of Mathematical Analysis and Applications》2007,329(1):51-64
This paper solves the following form of normalized eigenvalue problem: