Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

Bounds for the Steklov eigenvalues

Let \(\Omega \) be a star-shaped bounded domain in \((\mathbb {S}^{n}, ds^{2})\) with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in \(\Omega .\) This result extends a result given by Kuttler and Sigillito (SIAM Rev 10:368–370, 1968) for a star-shaped bounded domain in \(\mathbb {R}^2\). Further we also obtain a two sided bound for the eigenvalues of the Steklov problem on a ball in \(\mathbb {R}^n\) with rotationally invariant metric and with bounded radial curvature.     

Bounds for eigenvalues of singular differential operators

The spectrum of the Sturm-Liouville operator is studied, and the principal results consist of two-sided bounds of the eigenvalue distribution and of a criterion establishing that the resolvent belongs to a symmetrically normed ideal .Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 859–867, December, 1976.In conclusion the author expresses his gratitude to M. Z. Solomyak for valuable critical remarks.     

Bounds for ratios of the membrane eigenvalues

For a bounded planar region in R², we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k+1)-th (k?3) membrane eigenvalues.     

Bounds for eigenvalues of certain stochastic matrices

We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦?1–n^-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.     

Bounds on distances between eigenvalues

Explicit (computable) lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining (not necessarily real) eigenvalues of the matrix are developed.     

Bounds for eigenvalues of second-order elliptic differential operators

Summary We derive bounds for the firstN eigenvalues of a linear second-order elliptic differential operator on a bounded domain, subject to mixed boundary conditions. The results are achieved by a combination of (a generalized version of) Kato's estimates and a homotopy algorithm.     

Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R ⁿ . If λ _k+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and and, for any n and with , where j _p,k denotes the k-th positive zero of the standard Bessel function J _p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceH. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA     

Bounds of eigenvalues of a graph

LetG be a simple graph withn vertices. We denote by _i(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that –1₂(G)(n–2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG2K _n/2.     

Bounds on eigenvalues and chromatic numbers

We give new bounds on eigenvalue of graphs which imply some known bounds. In particular, if T(G) is the maximum sum of degrees of vertices adjacent to a vertex in a graph G, the largest eigenvalue ρ(G) of G satisfies with equality if and only if either G is regular or G is bipartite and such that all vertices in the same part have the same degree. Consequently, we prove that the chromatic number of G is at most with equality if and only if G is an odd cycle or a complete graph, which implies Brook's theorem. A generalization of this result is also given.     

Bounds for the real eigenvalues of a cascade matrix

Bounds are derived for the real eigenvalues of a special matrix. Matrices of this form arise in the design of two-up one-down cascades for isotope separation.     

Bounds for eigenvalues and singular values of matrix completions

Bounds are given for eigenvalues of hermitian completions of partial hermitian matrices, and for singular values of completions of partial block triangular matrices.     

Bounds for eigenvalues of symmetric block Jacobi scaled matrices

The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles. Translated by L. Yu. Kolotilina. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 18–25.     

Bounds for the stieltjes transform of spectral functions of singular eigenvalues

The main problem consists in obtaining G-estimates for singular eigenvalues of real matrices A=(a_ij),i = ,i = if the only things known are X_i, i = , independent observations over the matrix A+, where is a stochastic matrix. Under certain conditions on , A, n, m, and s results are obtained for the Stieltjes transform of spectral functions of the singular eigenvalues of the matrix A.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 464–469, April, 1990.     

Bounds on eigenvalues of Sturm-Liouville problems with discontinuous coefficients

This paper is concerned with obtaining upper and lower bounds for the eigenvalues of Sturm-Liouville problems with discontinuous coefficients. Such problems occur naturally in many areas of composite material mechanics.The problem is first transformed by using an analog of the classical Liouville transformation. Upper bounds are obtained by application of a Rayleigh-Ritz technique to the transformed problem. Explicit lower bounds in terms of the coefficients are established. Numerical examples illustrate the accuracy of the results.

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Résumé Dans cet article les bornes supérieures et inférieures sont détermineés pour les valeurs caractéristiques des problèmes de Sturm-Liouville avec des coefficients discontinus. De tels problèmes se trouvent naturellement dans la mécanique des materiaux composites.Après avoir transformé ce problème en utilisant un analogue de la transformation classique de Liouville, les bornes supérieures sont obtenues par l'application d'une technique de Rayleigh-Ritz au problème transformé. Les bornes inférieurs sont determinées en fonction des coefficients sous une forme explicite. Quelques exemples numériques montrent l'exactitude des résultats.

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This work was supported by the U.S. Army Research Office under Grants DAH C04-75-G-0059, DAAG 29-76-G-0063 and DAAG 29-77-G-0034.     

Bounds for the positive eigenvalues of the p-Laplacian with decaying potential

An upper bound is obtained for the positive eigenvalues of the p-Laplacian with decaying potential on [0,∞). The bound is expressed in terms of the potential and is shown to be the best possible of its kind.