190509_QE_QG_EQ.pdf (511.21 kB)
Quantum ergodicity for large equilateral quantum graphs
journal contribution
posted on 2019-06-26, 10:10 authored by Maxime Ingremeau, Mostafa Sabri, Brian WinnBrian WinnConsider 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 SocietyVolume
101Issue
1Pages
82 - 109Citation
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
WileyVersion
- AM (Accepted Manuscript)
Rights holder
© London Mathematical SocietyPublisher 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
2019-06-07Publication date
2019-07-26Copyright date
2019ISSN
0024-6107Publisher version
Language
- en