Quantum diffusion of the random Schrödinger evolution in the scaling limit

L. Erdös, M. Salmhofer, H. Yau, Acta Mathematica 200 (2008) 211–277.

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Abstract
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.
Publishing Year
Date Published
2008-07-01
Journal Title
Acta Mathematica
Volume
200
Issue
2
Page
211 - 277
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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
Erdös, L., Salmhofer, M., & Yau, H. (2008). Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Mathematica, 200(2), 211–277. https://doi.org/10.1007/s11511-008-0027-2
Erdös, László, Manfred Salmhofer, and Horng Yau. “Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit.” Acta Mathematica 200, no. 2 (2008): 211–77. https://doi.org/10.1007/s11511-008-0027-2.
L. Erdös, M. Salmhofer, and H. Yau, “Quantum diffusion of the random Schrödinger evolution in the scaling limit,” Acta Mathematica, vol. 200, no. 2, pp. 211–277, 2008.
Erdös L, Salmhofer M, Yau H. 2008. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Mathematica. 200(2), 211–277.
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.

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