Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Eigenvalues of collision operators: Properties and methods of computation

Abstract: The linear Boltzmann equation for active atoms submerged in the much denser perturber gas contains a collision rate and a kernel. These two quantities are combined into a single entity\char22{}the collision operator. The collision operator possesses several interesting properties, the most important being that it is Hermitian. The eigenvalues are negative with the exception of one eigenvalue, which is zero and corresponds to the Maxwellian (steady-state) velocity distribution. A set of functions, closely related to the eigenfunctions of the quantum-mechanical harmonic oscillator, is postulated to approximate the true eigenfunctions. This assumption was a basis of the method of modeling various physical phenomena occurring in the gaseous mixtures, subjected to a radiation field. The eigenvalues of the collision operator were treated as free parameters. In this paper we establish a direct relationship between the eigenvalues and the collision integrals, or transport coefficients, known from the kinetic theory of gases. The generating function approach is employed to derive expressions yielding the eigenvalues. The obtained results form a bridge between kinetic theory, atomic physics, and quantum optics.

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

Abstract: We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\pm \infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.

A two points connection problem involving logarithmic polynomials

Abstract: t— = (A + tB)X dt was studied under the assumptions that the eigenvalues of the diagonal matrix B satisfy the pentagonal condition, and that the matrix A has no congruent eigenvalues, or only one pair of congruent eigenvalues. In this paper, we extend these results in the direction that the matrix A may have any sets of congruent eigenvalues. Although we will investigate the case where all the eigenvalues of the matrix A are congruent for the sake of simplicity, the method applies easily to the general case to yield the similar results. A system of n linear ordinary differential equations of the form (1.1) has a regular singular point at t = 0, and an irregular singular point of rank one at t = oo.

On the Finiteness of the Discrete Spectrum of a Four-Particle Lattice Schrödinger Operator

Abstract: A Hamiltonian describing four bosons that move on a lattice and interact by means of pair zero-range attractive potentials is considered. A stronger version of the Hunziker–Van Vinter–Zhislin theorem on the essential spectrum is established. It is proved that the set of eigenvalues lying to the left of the essential spectrum is finite for any interaction energy of two bosons and is empty if this energy is sufficiently small.

Dependence of eigenvalues of 2nth order boundary value transmission problems

Abstract: The present paper deals with the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. The eigenvalues depend not only continuously but also smoothly on the problem. Some new differential expressions of eigenvalues with respect to an endpoint, a coefficient, the weight function, boundary conditions, and transmission conditions, are given.