Quantum diffusion of the random Schrödinger evolution in the scaling limit
László Erdös
Salmhofer, Manfred
Yau, Horng-Tzer
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.
Springer
2008
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doc-type:article
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http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/2753
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>
info:eu-repo/semantics/altIdentifier/doi/10.1007/s11511-008-0027-2
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