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Quantum ergodicity for large equilateral quantum graphs

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journal contribution
posted on 26.06.2019, 10:10 by Maxime Ingremeau, Mostafa Sabri, Brian WinnBrian Winn
Consider a sequence of finite regular graphs converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero coupling constant α) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum, the eigenfunctions of the converging quantum graphs satisfy a quantum ergodicity theorem. In case α = 0 and U = 0, the limit measure is the uniform measure on the edges. In general, it has an explicit C 1 density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

Funding

Agence Nationale de laRecherche project GeRaSic (ANR-13-BS01-0007-01).

LabEx IRMIA.

Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of the London Mathematical Society

Volume

101

Issue

1

Pages

82 - 109

Citation

INGREMEAU, M., SABRI, M. and WINN, B., 2019. Quantum ergodicity for large equilateral quantum graphs. Journal of the London Mathematical Society, 101 (1), pp.82-109.

Publisher

Wiley

Version

AM (Accepted Manuscript)

Rights holder

© London Mathematical Society

Publisher statement

This is the peer reviewed version of the following article: INGREMEAU, M., SABRI, M. and WINN, B., 2019. Quantum ergodicity for large equilateral quantum graphs. Journal of the London Mathematical Society, 101 (1), pp.82-109, which has been published in final form at https://doi.org/10.1112/jlms.12259. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.

Acceptance date

07/06/2019

Publication date

2019-07-26

Copyright date

2019

ISSN

0024-6107

Language

en

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