Towards the Quantum Brownian Motion

Abstract

We consider random Schr\"odinger equations on $\bR^d$ or $\bZ^d$ for d≥3 with uncorrelated, identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. Suppose that the space and time variables scale as x∼λ−2−κ/2,t∼λ−2−κ with 0<κ≤κ0, where κ0 is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of ψt, $\bE W_{\psi_{t}} (x, v)$, converges weakly to a solution of a heat equation in the space variable x for arbitrary L2 initial data in the weak coupling limit λ→0. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum v.