High Energy Asymptotics and Trace Formulas for the Perturbed Harmonic Oscillator

A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λ_n, a complete asymptotic expansion for large n is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularized sums of integer powers of eigenvalues λ_n in terms of the heat invariants. Communicated by Bernard Helffer submitted 15/06/05, accepted 16/09/05     

The condition numbers of the matrix eigenvalue problem

Summary For ann ×n matrixA with distinct eigenvalues explicit expressions are obtained for certain condition numbers associated with the reduction ofA to its Jordan normal form. These condition numbers are also related by inequalities to (i) the departure from normality ofA, (ii) the discriminant of the eigenvalues ofA, (iii) the Gram determinant of the eigenvectors ofA.     

Simultaneous estimation of eigenvalues

The problem of simultaneous estimation of eigenvalues of covariance matrix is considered for one and two sample problems under a sum of squared error loss. New classes of estimators are obtained which dominate the best multiple of the sample eigenvalues in terms of risk. These estimators shrink or expand the sample eigenvalues towards their geometric mean. Similar results are obtained for the estimation of eigenvalues of the precision matrix and the residual matrix when the original covariance matrix is partitioned into two groups. As a consequence, a new estimator of trace of the covariance matrix is obtained.The results are extended to two sample problem where two Wishart distributions are independently observed, say, S _i W _p ( _i , k _i ), i=1, 2, and eigenvalues of _{1 2} ^-1 are estimated simultaneously. Finally, some numerical calculations are done to obtain the amount of risk improvement.     

Limiting Absorption Principle for Stratified Operators with Thresholds: A General Method

Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z ∈ I. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).     

The Laplacian spread of graphs

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c-cyclic graphs with n vertices and Laplacian spread n − 1 are discussed.     

Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators

We develop a new method for obtaining bounds on the negative eigenvalues of self-adjoint operators B in terms of a Schatten norm of the difference of the semigroups generated by A and B, where A is an operator with non-negative spectrum. Our method is based on the application of the Jensen identity of complex function theory to a suitably constructed holomorphic function, whose zeros are in one-to-one correspondence with the negative eigenvalues of B. Applying our abstract results, together with bounds on Schatten norms of semigroup differences obtained by Demuth and Van Casteren, to Schr?dinger operators, we obtain inequalities on moments of the sequence of negative eigenvalues, which are different from the Lieb–Thirring inequalities. Guy Katriel: Partially supported by the Minerva Foundation (Germany). Submitted: September 4, 2007. Accepted: December 11, 2007.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.     

On a class of quasilinear eigenvalue problems in RN

We study an eigenvalue problem in R ^N which involves the p ‐Laplacian (p > N ≥ 2) and the nonlinear term has a global (p – 1)‐sublinear growth. The existence of certain open intervals of eigenvalues is guaranteed for which the eigenvalue problem has two nonzero, radially symmetric solutions. Some stability properties of solutions with respect to the eigenvalues are also obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Posterior two-sided estimates for eigenvalues of the singular Sturm-Liouville problem

Based on three-point difference and variational-difference schemes for auxiliary nonsingular spectral problems providing for a two-sided approximation of eigenvalues of the singular Sturm-Liouville problem, posterior upper and lower estimates for eigenvalues of the input singular problem are obtained. Bibliography: 3 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 39–49.     

On the asymptotic error estimate for a discrete problem of fourth-order accuracy

The leading term of the error of eigenvalues of a discrete analog of the eigenvalue problem for an elliptic operator with variable coefficients is obtained. A method for refining eigenvalues by evaluating a correction with the help of a discrete problem of second-order accuracy is proposed. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 153–160.     

A sufficient condition that a matrix have eigenvalues with non-zero real parts

If A is a complex matrix let à be the real matrix obtained by replacing the diagonal elements of A by the moduli of their real parts and by replacing the off-diagonal elements by the negative of their moduli. Then we show that if à is an M-matrix the eigenvalues of A have nonzero real parts and, moreover, the moduli of the real parts are bounded below by the minimum of the real parts of the eigenvalues of Ã.     

On complex eigenvalues ofM andP matrices

Summary Inequalities are obtained for the complex eigenvalues of anM matrix or aP matrix which depend only on the order of the matrix.This research was supported in part by the National Science Foundation under Grant GP-20555.     

The zeta functions of complexes from PGL(3): A representation-theoretic approach

The zeta function attached to a finite complex X _Γ arising from the Bruhat-Tits building for PGL₃(F) was studied in [KL], where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of X _Γ. In this paper we re-establish the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.     

Inverse nodal problem for p–laplacian dirac system

In this study, we solve an inverse nodal problem for p‐Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions r(x) and q(x) in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.     

A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix

Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required.The technique is also applicable to other computational methods which use the bisection process.     

Spectra of Some Interesting Combinatorial Matrices Related to Oriented Spanning Trees on a Directed Graph

The Laplacian of a directed graph G is the matrix L(G) = O(G) –, A(G) where A(G) is the adjaceney matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) and their multiplicities in terms of the eigenvalues of the induced subgraphs and the Laplacian matrix of G. Finally we compute the eigenvalues of D(G) for some specific directed graphs G. A recent conjecture of Propp for D(H _n ) follows, where H _n stands for the complete directed graph on n vertices without loops.     

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.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right‐hand sides. These preconditioners based on limited memory quasi‐Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first‐level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half‐plane, respectively. Third, we focus on theoretical properties of the Ritz‐LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large‐scale symmetric saddle‐point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications. Copyright © 2016 John Wiley & Sons, Ltd.