Quantum ergodicity for periodic graphs

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.