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res:
bibo_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.
@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: László
foaf_name: László Erdös
foaf_surname: Erdös
foaf_workInfoHomepage: http://www.librecat.org/personId=4DBD5372-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0001-5366-9603
- foaf_Person:
foaf_givenName: Manfred
foaf_name: Salmhofer, Manfred
foaf_surname: Salmhofer
- foaf_Person:
foaf_givenName: Horng
foaf_name: Yau, Horng-Tzer
foaf_surname: Yau
bibo_doi: 10.1007/s11511-008-0027-2
bibo_issue: '2'
bibo_volume: 200
dct_date: 2008^xs_gYear
dct_publisher: Springer@
dct_title: Quantum diffusion of the random Schrödinger evolution in the scaling
limit@
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