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Abstract:
The phenomenon of wave localization permeates 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 known methods predicted specific spatial location or frequencies of the localized waves.
In this talk I will present recent results revealing a universal mechanism of localization for low-energy eigenfunctions, applying to boundary problems for the Laplacian and bilaplacian, Schroedinger operator with disordered potential, $div A\nabla$, and any other elliptic operator 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 sharp 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 notoriously fail.
A substantial portion of the talk will be devoted to impact of the geometry (of the original domain and the newly discovered localization subregions) on the properties of solutions. We will, in particular, discuss dimension and absolute continuity of harmonic measure, rectifiability, and related issues.
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