# A geometric method for eigenvalue problems with low rank perturbations

## TL;DR

This paper introduces a geometric approach leveraging differential geometry to analyze the spectrum of operators with low-rank non-normal perturbations, applicable to various models in applied mathematics.

## Contribution

It presents a novel geometric method for explicitly determining spectra of low-rank perturbed operators, extending classical techniques to non-normal cases.

## Key findings

- Successfully analyzed three applied models using the geometric method.
- Provided explicit spectral characterizations for low-rank perturbations.
- Demonstrated the method's effectiveness in non-normal operator contexts.

## 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).