Article

Annales Henri Poincaré

, Volume 8, Issue 4, pp 621-685

Open Access This content is freely available online to anyone, anywhere at any time.

Quantum Diffusion for the Anderson Model in the Scaling Limit

  • László ErdősAffiliated withInstitute of Mathematics, University of Munich Email author 
  • , Manfred SalmhoferAffiliated withMax-Planck Institute for MathematicsTheoretical Physics, University of Leipzig
  • , Horng-Tzer YauAffiliated withDepartment of Mathematics, Harvard University

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.