A geometric method for eigenvalue problems with low-rank perturbations

Thomas J Anastasio; Andrea K Barreiro; Jared C Bronski

doi:10.1098/rsos.170390

A geometric method for eigenvalue problems with low-rank perturbations

R Soc Open Sci. 2017 Sep 27;4(9):170390. doi: 10.1098/rsos.170390. eCollection 2017 Sep.

Authors

Thomas J Anastasio¹, Andrea K Barreiro², Jared C Bronski³

Affiliations

¹ Department of Molecular and Integrative Physiology and Beckman Institute, University of Illinois Urbana-Champaign, Urbana, IL 61820, USA.
² Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX 75275, USA.
³ Department of Mathematics, University of Illinois Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA.

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 analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci.22, 239-254. (doi:10.1007/s10827-006-0010-x)), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math.48, 249-264. (doi:10.1093/imamat/48.3.249)).

Keywords: Aronszajn–Krein formula; bifurcation theory; rank-one perturbations.