A geometric method for eigenvalue problems with low-rank perturbations

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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)).

Original languageEnglish (US)
Article number170390
JournalRoyal Society Open Science
Issue number9
StatePublished - Sep 27 2017


  • Aronszajn-Krein formula
  • Bifurcation theory
  • Rank-one perturbations

ASJC Scopus subject areas

  • General


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