A geometric method for eigenvalue problems with low rank perturbations

Abstract

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyze the spectrum. We use these techniques to analyze three problems of this form: a model of the oculomotor integrator due to Anastasio and Gad (2007), a continuum integrator model, and a nonlocal model of phase separation due to Rubinstein and Sternberg (1992).