Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Estimates for the real and imaginary parts of the eigenvalues of matrices and applications

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.     

Finitely smooth normal form of an autonomous system with two pure imaginary roots

We consider the problem of finitely smooth normalization of a system of ordinary differential equations whose linear part has two eigenvalues, while the other eigenvalues lie outside the imaginary axis.     

On the distribution of Laplacian eigenvalues of trees

For a tree T$T$ with n$n$ vertices, we apply an algorithm due to Jacobs and Trevisan (2011) to study how the number of small Laplacian eigenvalues behaves when the tree is transformed by a transformation defined by Mohar (2007). This allows us to obtain a new bound for the number of eigenvalues that are smaller than 2. We also report our progress towards a conjecture on the number of eigenvalues that are smaller than the average degree.     

Limit points of eigenvalues of (di)graphs

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D, the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set D of digraphs (graphs) is a limit point of eigenvalues of a set of bipartite digraphs (graphs), where consists of the double covers of the members in D. 4. Every limit point of eigenvalues of a set D of digraphs is a limit point of eigenvalues of line digraphs of the digraphs in D. 5. If M is a limit point of the largest eigenvalues of graphs, then −M is a limit point of the smallest eigenvalues of graphs.     

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

An important step in estimating the index of refraction of electromagnetic scattering problems is to compute the associated transmission eigenvalue problem. We develop in this paper efficient and accurate spectral methods for computing the transmission eigenvalues associated to the electromagnetic scattering problems. We present ample numerical results to show that our methods are very effective for computing transmission eigenvalues (particularly for computing the smallest eigenvalue), and together with the linear sampling method, provide an efficient way to estimate the index of refraction of a non-absorbing inhomogeneous medium.     

Expression for the number of eigenvalues of a Friedrichs model

We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues.     

Isoperimetric control of the Steklov spectrum

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

Absence of Critical Mass Densities for a Vibrating Membrane

We study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.     

On the asymptotic behavior of eigenvalues of the radial p-laplacian

We study the asymptotic behavior and distribution of the eigenvalues of the singular radial p-laplacian. We prove a Weyl type asymptotic formula for the number of eigenvalues less than a given value.     

具有转移条件的向量型Sturm-Liouville问题的特征值重数及Ambarzumyan定理

研究了定义在有限区间内具有转移条件的m维向量型Sturm-Liouville问题.主要得到了该问题特征值重数的若干结论.证明了当矩阵值势函数Q满足一定的条件时,只能有有限个重数为m的特征值.作为重数结果的应用,证明了该问题的Ambarzumyan定理.     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with modulus ρ(A) have the same geometric multiplicity. We also prove that twodimensional nonnegative tensors’ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.     

Eigenvalues of large sample covariance matrices of spiked population models

We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.     

Spectra of positive and negative energies in the linearized NLS problem

We study the spectrum of the linearized NLS equation in three dimensions in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. The nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum. We derive bounds on the number of unstable eigenvalues of the linearized NLS problem and study bifurcations of embedded eigenvalues of positive and negative energies. We develop the L²‐scattering theory for the linearized NLS operators and recover results of Grillakis [5] with a Fermi golden rule. © 2004 Wiley Periodicals, Inc.     

On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

Dirac Eigenvalues for Generic Metrics on Three-Manifolds

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.     

Some Spectral Properties of Metapositive Definite Matrice

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Distribution of eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues are infinitely dispersed and its application to minimax estimation of covariance matrix

We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually independently distributed. The limiting distributions of the normalized sample eigenvalues are chi-squared distributions with varying degrees of freedom and the distribution of the relevant elements of the eigenvectors is the standard normal distribution. As an application of this result, we investigate tail minimaxity in the estimation of the population covariance matrix of Wishart distribution with respect to Stein's loss function and the quadratic loss function. Under mild regularity conditions, we show that the behavior of a broad class of tail minimax estimators is identical when the sample eigenvalues become infinitely dispersed.