Using wavepackets to reach the diffusive limit of the Schrodinger equation in the weak coupling limit
The weak coupling limit of the random Schrodinger equation is a model for quantum particles moving in a random environment. In 2008, Erdos, Salmhofer, and Yau demonstrated that such particles exhibit diffusive behavior by deriving an effective heat-type equation for the Wigner transform of the wavefunctions. The Erdos-Salmhofer-Yau argument used the Fourier transform to compute the terms of a Duhamel expansion for the solution. In this talk I present an alternative approach using wavepackets, which gives the terms of the Duhamel expansion a geometric meaning in terms of an integral over piecewise linear paths. I will show how to use geometric arguments involving these piecewise linear paths to recover the kinetic limit of the solutions and also briefly sketch some of the ideas needed to reach the diffusive time scale.