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Quantum Ergodicity for Periodic Graphs

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Abstract

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.

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Notes

  1. It is worthwhile to note that in the case of trees [1, 3, 4], we usually evolve the dynamical system in time T, essentially up to the girth of the graph, take the size of the graph \(N\rightarrow \infty \), then finally take \(T\rightarrow \infty \). Here we first consider the equilibrium limit in T, then take \(N\rightarrow \infty \) in the end of the proof.

  2. Note that we only discussed the (ir)-reducibility of the Bloch variety here. The irreducibility of the Fermi variety, where \(\lambda \) is fixed, is significantly harder to prove [22], but we do not need it.

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Acknowledgements

The authors are thankful to Nalini Anantharaman for helpful discussions. M.S. is very thankful to Wencai Liu for discussions concerning his results [11, 12, 22, 23] on irreducibility of Bloch varieties.

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Authors and Affiliations

  1. Harvard University, Cambridge, MA, USA

    Theo McKenzie

  2. Cairo University, Giza, Egypt

    Mostafa Sabri

  3. New York University Abu Dhabi, Abu Dhabi, UAE

    Mostafa Sabri

  4. Stanford University, Stanford, CA, USA

    Theo McKenzie

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  1. Theo McKenzie
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Correspondence to Theo McKenzie.

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Communicated by J. Ding.

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T. McKenzie: Supported by NSF GRFP Grant DGE-1752814 and NSF Grant DMS-2212881.

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McKenzie, T., Sabri, M. Quantum Ergodicity for Periodic Graphs. Commun. Math. Phys. 403, 1477–1509 (2023). https://doi.org/10.1007/s00220-023-04826-2

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