Towards the quantum Brownian motion

We consider random Schr\"odinger equations on $\bR^d$ or $\bZ^d$ for $d\ge 3$ with uncorrelated, identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. Suppose that the space and time variables scale as $x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}$ with $0< \kappa \leq \kappa_0$, where $\kappa_0$ is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of $\psi_t$, $\bE W_{\psi_{t}} (x, v)$, converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data in the weak coupling limit $\lambda \to 0$. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$.