Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

TL;DR

This paper proves quantum ergodicity for expanding finite quantum graphs converging to spectrally delocalized infinite quantum trees, showing eigenfunctions are spatially delocalized in certain spectral intervals.

Contribution

It establishes quantum ergodicity for a new class of expanding quantum graphs converging to spectrally delocalized quantum trees, linking spectral properties to eigenfunction delocalization.

Findings

Eigenfunctions in the spectral interval I are spatially delocalized.

Quantum graphs exhibit quantum ergodicity under spectral delocalization assumptions.

Results connect spectral properties of quantum graphs to eigenfunction behavior.

Abstract

We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval $I$, in the sense that their spectrum in $I$ is purely absolutely continuous and their Green's functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in $I$ are spatially delocalized.