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A geometric method for analyzing operators with low-rank perturbations

Banff International Research Station for Mathematical Innovation and Discovery

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 self-adjoint operator. We use a simple idea of classical differential geometry (the envelope of a family of curves) to analyze the spectrum. When the rank of the perturbation is two, this allows us to view the system in a geometric way through a ``phase plan'' in the perturbation strengths. We show how to apply this technique to two problems: a neural network model of the oculomotor integrator (Anastasio and Gad 2007), and a nonlocal model of phase separation (Rubinstein and Sternberg 1992). This is work with Tom Anastasio and Jared Bronski (UIUC).

Mathematics; Partial differential equations; Dynamical systems and ergodic theory; Functional analysis

video/mp4

Author affiliation: Southern Methodist University

2017-12-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64053

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/