Quantum diffusion of the random Schrödinger evolution in the scaling limit

László Erdős Institute of Mathematics, University of Munich Manfred Salmhofer Max Planck Institute for Mathematics and Theoretical Physics, University of Leipzig Horng-Tzer Yau Department of Mathematics, Harvard University

TBD mathscidoc:1701.331994

Acta Mathematica, 200, (2), 211-277, 2006.4
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 $\psi_t$ 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$L$^{2}initial data.
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@inproceedings{lászló2006quantum,
  title={Quantum diffusion of the random Schrödinger evolution in the scaling limit},
  author={László Erdős, Manfred Salmhofer, and Horng-Tzer Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351955696703},
  booktitle={Acta Mathematica},
  volume={200},
  number={2},
  pages={211-277},
  year={2006},
}
László Erdős, Manfred Salmhofer, and Horng-Tzer Yau. Quantum diffusion of the random Schrödinger evolution in the scaling limit. 2006. Vol. 200. In Acta Mathematica. pp.211-277. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351955696703.
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