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Quantum ergodicity for large equilateral quantum graphs
Author(s) -
Ingremeau Maxime,
Sabri Mostafa,
Winn Brian
Publication year - 2020
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12259
Subject(s) - mathematics , quantum graph , equilateral triangle , eigenfunction , ergodicity , measure (data warehouse) , quantum , tree (set theory) , discrete mathematics , quantum mechanics , combinatorics , physics , eigenvalues and eigenvectors , geometry , graph , statistics , database , computer science
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.