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Quantum ergodicity for large equilateral quantum graphs
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2019-07-26 , DOI: 10.1112/jlms.12259
Maxime Ingremeau 1, 2 , Mostafa Sabri 3, 4 , Brian Winn 5
Affiliation  

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.

中文翻译:

大等边量子图的量子遍历性

考虑在Benjamini–Schramm的意义上收敛到无限规则树的一系列有限规则图。我们研究了等边边长,基尔霍夫条件(可能具有非零耦合常数)的诱导量子图 α )和对称势 ü 在边缘。我们证明在无限量子树具有绝对连续光谱的光谱区域中,会聚量子图的本征函数满足量子遍历定理。以防万一 α = 0 ü = 0 ,极限度量是边缘上的统一度量。一般来说,它有一个明确的 C 1个 密度。最后,我们证明了一个涉及积分算子的更强的量子遍历定理,其目的是研究本征函数相关性。
更新日期:2019-07-26
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