Quantum ergodicity for large equilateral quantum graphs

Maxime Ingremeau (1) , Mostafa Sabri (2) , Brian Winn (3)

1 IRMA - Institut de Recherche Mathématique Avancée
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Loughborough University

Consider a sequence of finite regular graphs (GN) 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 analytic density. We finally prove a stronger quantum ergodicity theorem involving integral operators, the purpose of which is to study eigenfunction correlations.

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Maxime Ingremeau, Mostafa Sabri, Brian Winn. Quantum ergodicity for large equilateral quantum graphs. J. London Math. Soc. (2), 2020, 101, pp.82-109. ⟨10.1112/jlms.12259⟩. ⟨hal-01735912⟩