{"abstract":[{"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&lt;ϰ&lt;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.\nThe 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 &gt; 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.\n","lang":"eng"}],"publication_status":"published","publication":"Acta Mathematica","issue":"2","quality_controlled":0,"title":"Quantum diffusion of the random Schrödinger evolution in the scaling limit","_id":"2753","author":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"László Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Salmhofer","first_name":"Manfred","full_name":"Salmhofer, Manfred"},{"last_name":"Yau","first_name":"Horng","full_name":"Yau, Horng-Tzer"}],"doi":"10.1007/s11511-008-0027-2","intvolume":"       200","extern":1,"type":"journal_article","publist_id":"4139","date_updated":"2021-01-12T06:59:28Z","volume":200,"date_published":"2008-07-01T00:00:00Z","day":"01","year":"2008","status":"public","publisher":"Springer","page":"211 - 277","month":"07","date_created":"2018-12-11T11:59:25Z","citation":{"ieee":"L. Erdös, M. Salmhofer, and H. Yau, “Quantum diffusion of the random Schrödinger evolution in the scaling limit,” <i>Acta Mathematica</i>, vol. 200, no. 2. Springer, pp. 211–277, 2008.","short":"L. Erdös, M. Salmhofer, H. Yau, Acta Mathematica 200 (2008) 211–277.","ama":"Erdös L, Salmhofer M, Yau H. Quantum diffusion of the random Schrödinger evolution in the scaling limit. <i>Acta Mathematica</i>. 2008;200(2):211-277. doi:<a href=\"https://doi.org/10.1007/s11511-008-0027-2\">10.1007/s11511-008-0027-2</a>","mla":"Erdös, László, et al. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit.” <i>Acta Mathematica</i>, vol. 200, no. 2, Springer, 2008, pp. 211–77, doi:<a href=\"https://doi.org/10.1007/s11511-008-0027-2\">10.1007/s11511-008-0027-2</a>.","apa":"Erdös, L., Salmhofer, M., &#38; Yau, H. (2008). Quantum diffusion of the random Schrödinger evolution in the scaling limit. <i>Acta Mathematica</i>. Springer. <a href=\"https://doi.org/10.1007/s11511-008-0027-2\">https://doi.org/10.1007/s11511-008-0027-2</a>","chicago":"Erdös, László, Manfred Salmhofer, and Horng Yau. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit.” <i>Acta Mathematica</i>. Springer, 2008. <a href=\"https://doi.org/10.1007/s11511-008-0027-2\">https://doi.org/10.1007/s11511-008-0027-2</a>.","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."}}