Quantum Ergodicity for Periodic Graphs

This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger 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 Schrödinger operators on $$\mathbb {Z}^d$$ . The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.