$L^p$ norms and support of eigenfunctions on graphs

TL;DR

This paper investigates the delocalization and support properties of eigenfunctions of Schrödinger operators on large finite graphs, providing bounds on their support size and Lp-norms across various graph sequences.

Contribution

It offers new bounds on eigenfunction support and Lp-norms for Schrödinger operators on large, possibly irregular graphs, including sequences like N-lifts.

Findings

Eigenfunctions have large support in prescribed energy regions.

Lp-norm estimates are established for eigenfunctions.

Results apply to irregular graphs and graph sequences such as N-lifts.

Abstract

This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our estimates hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts.