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Texas A&M University
Hyperbolic Soccer Ball
Hyperbolic soccer ball
Source: Tina Mai

Past Ewing Lectures

2019: Douglas N. Arnold

Wave localization and its landscape

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The puzzling phenomenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating media. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials.

Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential.

We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.