Towards the quantum Brownian motion

Abstract

We consider random Schrödinger 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 $0<\kappa \le {\kappa }_0$, where ${\kappa }_0$ is a sufficiently small universal constant. We prove that the expectation value of the Wigner distribution of ${\psi }_t$, $\bE W_{ψ_{t}} (x, v)$, 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 $v$.

Self-contained overview (Conference proceedings). The complete proof is archived in math-ph/0502025. Some typos corrected and new references added in the updated version