Voir la notice de l'article provenant de la source Annals of Mathematics website
We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that an absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply “quantum ergodicity,” a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies, in particular, to graphs converging to the Anderson model on a regular tree, in the regime of extended states studied by Klein and Aizenman-Warzel.
Nalini Anantharaman 1 ; Mostafa Sabri 2
@article{10_4007_annals_2019_189_3_3,
author = {Nalini Anantharaman and Mostafa Sabri},
title = {Quantum ergodicity on graphs: {From} spectral to spatial delocalization},
journal = {Annals of mathematics},
pages = {753--835},
publisher = {mathdoc},
volume = {189},
number = {3},
year = {2019},
doi = {10.4007/annals.2019.189.3.3},
mrnumber = {3961083},
zbl = {07097490},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.3/}
}
TY - JOUR AU - Nalini Anantharaman AU - Mostafa Sabri TI - Quantum ergodicity on graphs: From spectral to spatial delocalization JO - Annals of mathematics PY - 2019 SP - 753 EP - 835 VL - 189 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.3/ DO - 10.4007/annals.2019.189.3.3 LA - en ID - 10_4007_annals_2019_189_3_3 ER -
%0 Journal Article %A Nalini Anantharaman %A Mostafa Sabri %T Quantum ergodicity on graphs: From spectral to spatial delocalization %J Annals of mathematics %D 2019 %P 753-835 %V 189 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.3.3/ %R 10.4007/annals.2019.189.3.3 %G en %F 10_4007_annals_2019_189_3_3
Nalini Anantharaman; Mostafa Sabri. Quantum ergodicity on graphs: From spectral to spatial delocalization. Annals of mathematics, Tome 189 (2019) no. 3, pp. 753-835. doi: 10.4007/annals.2019.189.3.3
Cité par Sources :