Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams

L. Erdös, M. Salmhofer, H. Yau, Communications in Mathematical Physics 271 (2007) 1–53.

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We consider random Schrödinger equations on ℝd 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 χ ≃λ-2-κ/2, t ≃λ-2-κ with 0 < kappa; < kappa;0(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 L2 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.
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Date Published
2007-04-01
Journal Title
Communications in Mathematical Physics
Volume
271
Issue
1
Page
1 - 53
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Erdös L, Salmhofer M, Yau H. Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Communications in Mathematical Physics. 2007;271(1):1-53. doi:10.1007/s00220-006-0158-2
Erdös, L., Salmhofer, M., & Yau, H. (2007). Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Communications in Mathematical Physics, 271(1), 1–53. https://doi.org/10.1007/s00220-006-0158-2
Erdös, László, Manfred Salmhofer, and Horng Yau. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams.” Communications in Mathematical Physics 271, no. 1 (2007): 1–53. https://doi.org/10.1007/s00220-006-0158-2.
L. Erdös, M. Salmhofer, and H. Yau, “Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams,” Communications in Mathematical Physics, vol. 271, no. 1, pp. 1–53, 2007.
Erdös L, Salmhofer M, Yau H. 2007. Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams. Communications in Mathematical Physics. 271(1), 1–53.
Erdös, László, et al. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams.” Communications in Mathematical Physics, vol. 271, no. 1, Springer, 2007, pp. 1–53, doi:10.1007/s00220-006-0158-2.

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