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The Schrödinger equation with a random potential serves as a simple model for the propagation of waves in a disordered medium. It is conjectured that when the strength of the potential is weak, the solution should evolve diffusively as time goes to infinity. Previous approaches to this problem have hinged on diagrammatic expansions for the propagator. In this talk, I will explain a new proof of this phenomenon for long but finite times that instead proceeds by studying certain self-consistent equations for the resolvent. The analysis requires techniques from random matrix theory as well as dispersive properties of the Laplacian. This is joint work with Reuben Drogin and Felipe Hernández.

TBA