Quantum Ergodicity on Graphs : from Spectral to Spatial Delocalization

Abstract : 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 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 régime of extended states studied by Klein and Aizenman–Warzel.
Type de document :
Pré-publication, Document de travail
2017
Liste complète des métadonnées


https://hal.archives-ouvertes.fr/hal-01503951
Contributeur : Mostafa Sabri <>
Soumis le : vendredi 7 avril 2017 - 19:15:46
Dernière modification le : jeudi 13 avril 2017 - 01:03:34

Fichiers

170405QE_det.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01503951, version 1
  • ARXIV : 1704.02766

Collections

Citation

Nalini Anantharaman, Mostafa Sabri. Quantum Ergodicity on Graphs : from Spectral to Spatial Delocalization. 2017. <hal-01503951>

Partager

Métriques

Consultations de
la notice

93

Téléchargements du document

27