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