Eigenvalue Problem For Differential Operator Research Articles

Mathematical analysis of bent optical waveguide eigenvalue problem

Abstract This work investigates a mathematical model of the propagation of lightwaves in bent optical waveguides. This modeling leads to a non-self-adjoint eigenvalue problem for differential operator defined on the unbounded domain. Through detailed analysis, a relationship between the real and imaginary parts of the complex-valued propagation constants was constructed. Using this relation, it is found that the real and imaginary parts of the propagation constants are bounded, meaning they are limited within certain region in the complex plane. The orthogonality of these bent modes is also proved. By the asymptotic analysis of these modes, it is proved that as r → ∞ the behavior of the eigenfunctions is proportional to 1 / r .

Finding Eigenvalues of Holomorphic Fredholm Operator Pencils Using Boundary Value Problems and Contour Integrals

Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators (and the associated eigenfunctions) via contour integrals of solutions to resolvent equations. The approach is based on Keldysh’ theorem and extends a recent method for matrices depending analytically on the eigenvalue parameter. We show that errors are well-controlled under very general assumptions when the resolvent equations are solved via boundary value problems on finite domains. Two applications are presented: an analytical study of Schrödinger operators on the real line as well as on bounded intervals and a numerical study of the FitzHugh–Nagumo system. We also relate the contour method to the well-known Evans function and show that our approach provides an alternative to evaluating and computing its zeros.

On the eigenvalue problems for differential operators with coupled boundary conditions

In the paper, the eigenvalue problems for one- and two-dimensional second order differential operators with nonlocal coupled boundary conditions are considered. Conditions for the existence of zero, positive, negative or complex eigenvalues are proposed and analytical expressions of eigenvalues are provided.

The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet– Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet–Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N + 1)-scale case.