DMS Applied Mathematics Seminar

Speaker: Joshua Lee Padgett, University of Arkansas

Title: Localization properties of discrete non-local Hamiltonian operators

Abstract: It is well known that certain physical systems may exhibit localized energy states in the presence of environmental disturbances. For numerous physically-relevant systems, this aforementioned phenomenon is known as Anderson localization. Anderson localization has attracted attention from the physics, mathematical physics, numerical analysis, and pure analysis communities, yet there are still many open questions related to the subject. In this talk we will provide a more operator-theoretical approach to the problem, which will provide two new directions of study for the Anderson localization problem. First, we will extend the problem to consider self-adjoint non-local operators on certain discrete graphs. Next, we will develop a novel method of studying the localization properties of these non-local operators via the consideration of their spectral properties. This approach allows for the development of surprising results that allow for the improvement of many existing results. This talk will include a review of the pertinent concepts from analysis, making the talk accessible to graduate students (even those who do not study pure analysis).