Quantum ergodicity for periodic graphs

TL;DR

This paper proves quantum ergodicity for a class of periodic Schr"odinger operators on large graphs, showing that most eigenfunctions become uniformly distributed, indicating delocalization.

Contribution

It establishes quantum ergodicity for periodic Schr"odinger operators on various lattices, extending previous results to a broader class of operators under Floquet eigenvalue assumptions.

Findings

Most eigenfunctions are equidistributed on large periodic graphs.

Results apply to adjacency matrices, triangular, honeycomb lattices, and Cartesian products.

Delocalization of eigenfunctions is demonstrated.

Abstract

We prove quantum ergodicity for a family of periodic Schr\"odinger operators $H$ on periodic graphs. This means that most eigenfunctions of $H$ on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on $\mathbb{Z}^d$, the triangular lattice, the honeycomb lattice, Cartesian products and periodic Schr\"odinger operators on $\mathbb{Z}^d$. The theorem applies more generally to any periodic Schr\"odinger operator satisfying an assumption on the Floquet eigenvalues.