Quantum ergodicity for large equilateral quantum graphs

TL;DR

This paper proves quantum ergodicity for large equilateral quantum graphs converging to an infinite regular tree, showing eigenfunction distribution becomes uniform or explicitly characterized in the limit.

Contribution

It establishes quantum ergodicity results for quantum graphs with equilateral edges and Kirchhoff conditions, extending understanding of eigenfunction behavior in the large graph limit.

Findings

Eigenfunctions become uniformly distributed on edges in the spectral regions with absolutely continuous spectrum.

Explicit analytic density describes the limit measure when potential and coupling are zero.

A stronger quantum ergodicity theorem involving integral operators is proved to analyze eigenfunction correlations.

Abstract

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 $\alpha$) 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 $\alpha$ = 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.