Tail bounds for the Dyson series of random Schrödinger equations

Schrödinger equations with weak random potentials are often studied via a perturbative expansion known as the Dyson series. In this talk, I will present probabilistic bounds for the terms of this series that, roughly speaking, demonstrate a “square-root cancellation” phenomenon induced by the randomness. I will briefly recall some motivation for these bounds, including applications to the delocalization of eigenfunctions. Then I will explain some elements of the proof, which combines tools from random matrix theory with dispersive properties of the Schrödinger equation. My intention is for this talk to be primarily expository, so I will provide much of the necessary background as we go. This is joint work with Reuben Drogin and Felipe Hernández.