Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

Algebraic conditions for <Emphasis Type="Italic">t</Emphasis>-tough graphs

We give some algebraic conditions for t-tough graphs in terms of the Laplacian eigenvalues and adjacency eigenvalues of graphs.     

Analytic continuation of eigenvalues of the Lamé operator

Eigenvalues of the Lamé operator are studied as complex-analytic functions in period τ of an elliptic function. We investigate the branching of eigenvalues numerically and clarify the relationship between the branching of eigenvalues and the convergent radius of a perturbation series.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

Exact <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L₂-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.Partially supported by grants of RFBR 01-01-00245 and 02-01-01099.     

Manifolds whose curvature operator has constant eigenvalues at the basepoint

Osserman conjectured that if the curvature operatorR of a Riemannian manifoldM has constant eigenvalues, thenM is locally a rank-1 symmetric space or is flat. The pointwise question is considerably more complicated. We present examples of Riemannian manifolds so thatR has constant eigenvalues at the basepoint, butR is not the curvature operator of a rank-1 symmetric space. Research partially supported by the NSF and IHES.     

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.     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be unbounded by making use of the Morse theory for aC ²-function at both isolated critical point and infinity.     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

First‐order systems in on with absolutely continuous real symmetric π‐periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet‐type boundary value problems on [0,π] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet‐type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright © 2014 John Wiley & Sons, Ltd.     

Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

In the first part of this paper we provide a self‐contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of C₀‐semigroups.     

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lamé eigenvalues with hard simple support boundary conditions does not depend on the Lamé coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lamé coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non‐convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non‐H² singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non‐convex polyhedron, the spectrum cannot be approximated by finite element methods using H¹ elements. Similar properties hold in polygons. We give numerical results for two L‐shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

Free membrane eigenvalues

For a simply connected and normalized domain D in the plane it was proven by Pólya and Schiffer in 1954 for the fixed membrane eigenvalues

Dark solitons in external potentials

We consider the persistence and stability of dark solitons in the Gross–Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton’s particle law for its position.      

On the spectral estimates for the Schrödinger operator on ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>, <Emphasis Type="Italic">d</Emphasis> ≽ 3

For the discrete Schr?dinger operator we obtain sharp estimates for the number of negative eigenvalues. Bibliography: 19 titles. To Nina N. Uraltseva, with admiration Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 107–126.     

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models.The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.     

加权流形上加权p-Laplace特征值问题的第一特征值下界估计

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.