Quantum Diffusion for the Anderson Model in the Scaling Limit

Abstract

We consider random Schrödinger equations on ℤdZd for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x∼λ−2−κ/2,t∼λ−2−κx∼λ−2−κ/2,t∼λ−2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to a solution of a heat equation in the space variable x for arbitrary L2 initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ.

This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved.