We introduce a new approach to the diffusive limit of the random Schrödinger
equation, first studied by L. Erdős, M. Salmhofer, and H.-T. Yau. Our approach is
based on a wavepacket decomposition of the evolution operator, which allows us to
interpret the Duhamel series as an integral over piecewise linear paths. We relate the
geometry of these paths to combinatorial features of a diagrammatic expansion which
allows us to express the error terms in the expansion as an integral over
paths that are exceptional in some way. These error terms are bounded using
geometric arguments. The main term is then shown to have a semigroup
property, which allows us to iteratively increase the timescale of validity of
an effective diffusion. This is the first derivation of an effective diffusion
equation from the random Schrödinger equation that is valid in dimensions
.
