Quantum ergodicity for periodic graphs
Abstract
We prove quantum ergodicity for a family of periodic Schr\"odinger operators on periodic graphs. This means that most eigenfunctions of on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on , the triangular lattice, the honeycomb lattice, Cartesian products and periodic Schr\"odinger operators on . The theorem applies more generally to any periodic Schr\"odinger operator satisfying an assumption on the Floquet eigenvalues.
Cite
@article{arxiv.2208.12685,
title = {Quantum ergodicity for periodic graphs},
author = {Theo Mckenzie and Mostafa Sabri},
journal= {arXiv preprint arXiv:2208.12685},
year = {2022}
}
Comments
Two important updates. (1) Wencai Liu arXiv:2210.10532 has solved the open problem of v1, so quantum ergodicity holds for periodic operators on $\mathbb{Z}^d$ in all dimensions. (2) We now prove the Floquet assumption cannot be dropped and replaced by mere ac spectrum. More additions and stronger conclusions are featured. 26 pages, 5 figures