Related papers: Double, borderline, and extraordinary eigenvalues …
Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for…
A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such…
An upper bound for the number of distinct eigenvalues of a perturbed matrix has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate is the central result in Farrell's work and can be applied to estimate the number of…
We consider the 3-dimensional Stark operator perturbed by a complex-valued potential. We obtain an estimate for the number of eigenvalues of this operator as well as for the sum of imaginary parts of eigenvalues situated in the upper…
We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its…
A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical…
Exceptional points and double poles of the S matrix are both characterized by the coalescence of a pair of eigenvalues. In the first case, the coalescence causes a defect of the Hilbert space. In the second case, this is not so as shown in…
An eigenvalue $\lambda$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues…
When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate…
In this article, we study Steklov eigenvalues and mixed Steklov Neumann eigenvalues on a smooth bounded domain in $\mathbb{R}^{n}$, $n \geq 2$, having a spherical hole. We focus on two main results related to Steklov eigenvalues. First, we…
The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be…
We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter with "a negative number of poles". More precisely, we treat in a unified manner…
In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find…
We consider a class of two-dimensional Schr\"odinger operator with a singular interaction of the $\delta$ type and a fixed strength $\beta$ supported by an infinite family of concentric, equidistantly spaced circles, and discuss what…
This article is an expanded version of the plenary talk given by Evans Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace operators and Schr\"odinger operators with potentials containing curvature on certain regions of…
Let $G$ be a graph of order $n$ and let $\mu_{1}\left(G\right) \geq \cdots\geq\mu_{n}\left(G\right) $ be the eigenvalues of its adjacency matrix. This note studies eigenvalue problems of Nordhaus-Gaddum type. Let $\overline{G}$ be the…
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal…
An analytical perturbative method is suggested for solving the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions where {\psi} vanishes on an irregular closed curve. We can thus find the energy levels of a quantum…
We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient…
Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values…