MiS Preprint Repository

We consider random Schrödinger equations on ${\mathbb{R}}^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential.

Denote by $\lambda$ the coupling constant and ${\psi }_t$ the solution with initial data ${\psi }_0$. The space and time variables scale as $x\sim {\lambda }^{-2-\kappa /2},t\sim {\lambda }^{-2-\kappa }$ with $0<\kappa <{\kappa }_0(d)$. We prove that, in the limit $\lambda \to 0$, the expectation of the Wigner distribution of ${\psi }_t$ converges weakly to the solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data.

The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the {\it non-ladder} diagrams is smaller than their "naive size" by an extra ${\lambda }^c$ factor {\em per non-(anti)ladder vertex} for some $c>0$.

This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series. The analysis of the recollision graphs is given in the companion paper (L. Erdös, M. Salmhofer and H.-T. Yau, Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Submitted to Commun. Math. Phys.)