Numerical approaches for some Nonlinear Eigenvalue Problems

TL;DR

This paper reviews theoretical and numerical methods for solving nonlinear eigenvalue problems, focusing on spectral and finite difference approaches for non-self-adjoint operators, highlighting challenges and comparing results with theory.

Contribution

It introduces numerical techniques for nonlinear eigenvalue problems, especially for non-self-adjoint operators, and compares computational results with existing theoretical insights.

Findings

Spectral and finite difference methods effectively compute eigenvalues.

Eigenvalues of non-self-adjoint operators are highly unstable.

Numerical results align with theoretical predictions.

Abstract

In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given for the pseudospectra. Then we present the numerical methods and results obtained for eigenvalues computation with spectral methods and finite difference discretization, in infinite or bounded domains. Comparison with theoretical results is done. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.