Showing papers on "Spectrum of a matrix published in 1972"

On the eigenvalues of the first boundary value problem in unbounded domains

Abstract: This paper is devoted to the investigation of the spectrum of a polyharmonic operator in unbounded domains. The class of domains for which the spectrum of the corresponding first boundary value problem is discrete is examined. The classical asymptotic formula for eigenvalues is extended to the case of domains of finite volume. A two-sided bound for the distribution function of the eigenvalues is obtained in the general case. If the domain behaves sufficiently regularly at infinity, then the upper and lower bounds coincide in order. The results are new also for the Laplace operator. Bibliography: 13 items.

Perturbation of embedded eigenvalues

Abstract: In [4] a Weinstein-Aronszajn multiplicity theory for embedded eigenvalues arising from a certain type of \"resonance\" was developed. The results announced here continue the work of [4], and generalize results of [2] and [3] to embedded eigenvalues of arbitrary finite multiplicity m, and to perturbations of infinite rank. In particular, we are able to discuss certain operators of quantum mechanics. A notable feature of the case m > 1 is the appearance of Puiseux series for the resonances, in analogy to their appearance in the perturbation theory of isolated eigenvalues of nonselfadjoint operators [6, Chapters 2 and 7].

Eigenvalues of matrices with prescribed entries

Abstract: It is shown that there exists an /i-square matrix all whose eigenvalues and n-1 of whose entries are arbitrarily prescribed. This result generalizes a theorem of L. Mirsky. It is also shown that there exists an «-square matrix with some of its entries prescribed and with simple eigenvalues, provided that n of the nonprescribed entries lie on a diagonal or, alternatively, provided that the number of prescribed entries does not exceed In—2. A well-known result of L. Mirsky [3] states essentially that, given any 2n—1 complex numbers Ax, ■ • • , Xn, ax, • • • , an_x, there exists an nsquare matrix with eigenvalues A,, • • • , Xn and n—\ of its main diagonal entries equal to ax, • • ■ , an_x. Related results for matrices over general fields were also obtained by Farahat and Ledermann [1]. We show that the restriction of the n—\ prescribed entries to the main diagonal is unnecessary. We first investigate the conditions under which there exists a matrix with prescribed entries and simple eigenvalues. A position in a matrix in which some entries have been prescribed is said to be free, if there is no prescribed entry in that position. By a diagonal in an «-square matrix we mean a set of n positions no two of which are in the same row or in the same column; i.e., positions (i, o(i)), /=1, •• • , n, for some permutation a. Theorem 1. Let ax, • • • , ani_n be n2—nprescribed complex numbers and let (it,ft), /= 1, ■ • • , n2—n, be prescribed different positions in an n-square matrix, such that the n remaining free positions form a diagonal of the matrix. Then there exists an n-square matrix with simple eigenvalues and with the prescribed entries at in the prescribed positions (it,jt), t= 1, • • • , n2—n. The number n of free positions cannot, in general, be decreased. Received by the editors June 1, 1971. AMS 1969 subject classifications. Primary 1525.

Lower Bounds for Eigenvalues with Displacement of Essential Spectra

Abstract: New constructions of comparison operators for rigorous lower bounds to eigenvalues of a class of self-adjoint operators are presented. The formulation uses noncompact finite perturbations to displace eigenvalues and essential spectra, and leads to workable numerical procedures. These methods make possible for the first time lower bound calculations for the lower eigenvalues of the Schrodinger operators for atoms and ions having three or more electrons.

Inequalities connecting the eigenvalues of a hermitian matrix with the eigenvalues of complementary principal submatrices

Abstract: Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.

A simple method for finding the Jordan form of a matrix

Abstract: A conceptually simple algorithm is presented for determining the orders of the minor blocks in the Jordan canonical form of a given matrix whose eigenvalues are known.

Extremal Properties of Resonance Eigenvalues

Abstract: The minimax theorem for the eigenvalues of Hermitian operators is reviewed and its inverse (maxmini) theorem is derived. The connection between the Hylleraas-Undheim theorem (HUT) and the minimax theorem is clarified, and an extension of the HUT to compound resonance states is obtained. Thus, rigorous upper bounds on the resonance energies may be obtained without the explicit use of the exact open-channel projection operators. The exchange and rearrangement problems are also considered.

Spektrale Invarianz bei verallgemeinerten Eigenfunktionsentwicklungen

Abstract: Let T be a selfadjoint operator in a Gelfand triplet ϕ⊂H⊂ϕ′. Examples show that there can occur generalized eigenvalues of T which are not in the Hilbert space spectrum σ(T) of T. Moreover, the fuction d, which assigns to each real number s the dimension of the generalized eigenspace corresponding to s, can be essentially greater then the von Neumann multiplicity function of T. We therefore construct a new triplet Ψ⊂H⊂Ψ′, closely related to the given Gelfand triplet, according to which the set of generalized eigenvalues of T is contained in σ(T), and the function d essentially equals the von Neumann multiplicity function of T. Then, in particular, the closure of the set of generalized eigenvalues equals σ(T). The expansion theorems in ϕ⊂H⊂ϕ′ are transferred to Ψ⊂H⊂Ψ′.