--- _id: '2753' abstract: - lang: eng text: | We consider random Schrödinger equations on R d for d ≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0 . The space and time variables scale as x∼λ−2−ϰ/2 and t∼λ−2−ϰ with 0<ϰ<ϰ0(d) . We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt 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 non-ladder diagrams is smaller than their “naive size” by an extra λ c factor 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. author: - first_name: László full_name: László Erdös id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Manfred full_name: Salmhofer, Manfred last_name: Salmhofer - first_name: Horng full_name: Yau, Horng-Tzer last_name: Yau citation: ama: Erdös L, Salmhofer M, Yau H. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Mathematica. 2008;200(2):211-277. doi:10.1007/s11511-008-0027-2 apa: Erdös, L., Salmhofer, M., & Yau, H. (2008). Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Mathematica. Springer. https://doi.org/10.1007/s11511-008-0027-2 chicago: Erdös, László, Manfred Salmhofer, and Horng Yau. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit.” Acta Mathematica. Springer, 2008. https://doi.org/10.1007/s11511-008-0027-2. ieee: L. Erdös, M. Salmhofer, and H. Yau, “Quantum diffusion of the random Schrödinger evolution in the scaling limit,” Acta Mathematica, vol. 200, no. 2. Springer, pp. 211–277, 2008. ista: Erdös L, Salmhofer M, Yau H. 2008. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Mathematica. 200(2), 211–277. mla: Erdös, László, et al. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit.” Acta Mathematica, vol. 200, no. 2, Springer, 2008, pp. 211–77, doi:10.1007/s11511-008-0027-2. short: L. Erdös, M. Salmhofer, H. Yau, Acta Mathematica 200 (2008) 211–277. date_created: 2018-12-11T11:59:25Z date_published: 2008-07-01T00:00:00Z date_updated: 2021-01-12T06:59:28Z day: '01' doi: 10.1007/s11511-008-0027-2 extern: 1 intvolume: ' 200' issue: '2' month: '07' page: 211 - 277 publication: Acta Mathematica publication_status: published publisher: Springer publist_id: '4139' quality_controlled: 0 status: public title: Quantum diffusion of the random Schrödinger evolution in the scaling limit type: journal_article volume: 200 year: '2008' ...