Acta Mathematica

Quantum diffusion of the random Schrödinger evolution in the scaling limit

László Erdős, Manfred Salmhofer, and Horng-Tzer Yau

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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 $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $. We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable x for arbitrary L2 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 λc 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.


The first auhor was partially supported by NSF grant DMS-0307295 and MacArthur Fellowship. The third author was partially supported by NSF grant DMS-0200235 and EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027.

Article information

Acta Math., Volume 200, Number 2 (2008), 211-277.

Received: 21 April 2006
Revised: 2 April 2007
First available in Project Euclid: 31 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 81T18: Feynman diagrams 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general) 82C44: Dynamics of disordered systems (random Ising systems, etc.)

2008 © Institut Mittag-Leffler


Erdős, László; Salmhofer, Manfred; Yau, Horng-Tzer. Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200 (2008), no. 2, 211--277. doi:10.1007/s11511-008-0027-2.

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