Localization of waves in random resonant media

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Introduction and overall objective: This project fall under the theme of trapping and localizing waves. The goal of theproject is to show that waves travelling through an unbounded domain can be trapped if it is a composite (i.e.heterogeneous) medium with a special interplay between the microgeometry and the material properties. Moreprecisely, we will consider a space with two key features: (1) on an extended subset, the coefficients of the equationtake large values, and (2) a random landscape outside this subset.The topic of wave localization is of interest in the physics community because it is contrary to our everydayexperience, for instance, sound (acoustic waves) travel very far in open spaces. For this reason, this phenomenonreceived much attention when it first appeared in the quantum physics literature in the 1950s. However, amathematically rigorous study of this 'Anderson localization' phenomenon is only available for a few specific models,so the work is far from complete. Mathematically, we are working with a random environment. The novelty of theproject is to then consider adding in (1), with the expectation that the new setting exhibits 'stronger' wave trappingbehavior.Methodology and initial aims: Broadly, the subjects of this project include: probability theory, linear analysis,scattering theory, and mathematical physics. Concretely, this project will study the spectral properties of a randomdifferential operator. In terms of the setup, we will take one of the models that has been proven to exhibit Andersonlocalization behaviour, and modify it to have periodic regions with a coefficient that we will take to infinity. Themodification is to simulate (1). The initial aim is to prove localization for a fixed coefficient in (1) and a convenientchoice of probability measure in (2). In terms of the proof methods, we will look to modify either of the two mainmethods used to prove Anderson localization (multi-scale method, fractional moments m ethod.)