, Volume 8, Issue 4, pp 621-685,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 07 Jun 2007

Quantum Diffusion for the Anderson Model in the Scaling Limit

Abstract.

We consider random Schrödinger equations on \({\mathbb{Z}}^d\) for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ 0. The space and time variables scale as \(x \sim \lambda^{-2-\kappa/2},t \sim \lambda^{-2-\kappa}\) with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ t converges weakly to a solution of a heat equation in the space variable x for arbitrary L 2 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.

Communicated by Frank den Hollander.
Submitted: April 18, 2006. Accepted: October 12, 2006.
László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027.
Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship.
Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.