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The hidden landscape of localization of eigenfunctions

Recent Developments in Harmonic Analysis May 15, 2017 - May 19, 2017

May 18, 2017 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Svitlana Mayboroda (University of Minnesota, Twin Cities)
Location: SLMath: Eisenbud Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Abstract

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

 

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$,  and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.

Supplements
28692?type=thumb Mayboroda Notes 2.53 MB application/pdf Download