Communications in Mathematical Physics

, Volume 271, Issue 1, pp 1–53 | Cite as

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

  • László ErdősEmail author
  • Manfred Salmhofer
  • Horng-Tzer Yau


We consider random Schrödinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ0. 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 the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.


Anderson Model Feynman Graph Circle Graph Ladder Diagram Triple Collision 
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© Springer-Verlag 2007

Authors and Affiliations

  • László Erdős
    • 1
    Email author
  • Manfred Salmhofer
    • 2
    • 3
  • Horng-Tzer Yau
    • 4
  1. 1.Institute of MathematicsUniversity of MunichMunichGermany
  2. 2.Max–Planck Institute for MathematicsLeipzigGermany
  3. 3.Theoretical PhysicsUniversity of LeipzigLeipzigGermany
  4. 4.Department of MathematicsHarvard UniversityCambridgeUSA

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