Concerning the Spatial Localization of Eigenvectors (and Waves) in Complex Media

Jeffrey Ovall, Portland State University, Mathematics and Statistics SCAIM Seminar
February 7, 2023 12:00 pm ESB 4133

It is well-known that a function \(u=u(t,x)\) describing the behavior of acoustic or electromagnetic waves in time and space can often be decomposed as an infinite sum
u(t,x)=\sum_{n=1}^\infty c_n(t)\psi_n(x)~,
where each term in the sum is a product of a function \(c_n\) varying only in time and a function \(\psi_n\) varying only in space.  The standing waves \(\psi_n\) are eigenvectors of a spatial differential operator associated with the medium through which the waves are propagating. It is not as well-known that properties of the medium can cause some eigenvectors to be strongly spatially localized. A practical consequence of eigenvector localization is that waves at certain frequencies can be “trapped” at some location or “channelled” along some favorable path.  Such features are of interest in the design of structures having desired acoustic or electromagnetic properties: sound-mitigating outdoor barriers and next generation organic LEDs and solar cells are examples of this design principle in action. There remain many open problems related to understanding and exploiting this kind of localization, and we will discuss a computational approach that we hope will provide insight.  More specifically, we focus on the issue of eigenvector localization, outlining our computational approach and providing theoretical, heuristic, and empirical support for it through several examples (with many pictures).  We will also provide a (highly abridged) history of the study of localization.

Pizza lunch will be provided.

We acknowledge financial support from the Pacific Institute for the Mathematical Sciences (PIMS) and the UBC Institute of Applied Mathematics (IAM).