The phenomenon of wave localization permeates acoustics, quantum physics, energy engineering. It was used in the construction of noise abatement walls, LEDs, optical devices. Localization of quantum states of electrons by a disordered potential has become one of the prominent subjects in quantum physics, as well as harmonic analysis and probability. 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 an elliptic operator in 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. In the context of the Schrödinger operator, the landscape further provides sharp estimates on the exponential decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both semi-classical Agmon estimates and Weyl law notoriously fail.
This is joint work with D. Arnold, G. David, M. Filoche, and D. Jerison.