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On Spectral Bands of Discrete Periodic Operators

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Abstract

We consider discrete periodic operator on \({\mathbb {Z}}^d\) with respect to lattices \(\Gamma \subset {\mathbb {Z}}^d\) of full rank. We describe the class of lattices \(\Gamma \) for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed \(d-2\).

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Authors and Affiliations

  1. St. Petersburg Department of V. A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg, Russia, 191023

    Nikolay Filonov

  2. St. Petersburg State University, Universitetskaya emb. 7/9, St. Petersburg, Russia, 199034

    Nikolay Filonov

  3. Department of Mathematics, Michigan State University, Wells Hall, 619 Red Cedar Road, East Lansing, MI, 48910, USA

    Ilya Kachkovskiy

Authors
  1. Nikolay Filonov
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  2. Ilya Kachkovskiy
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Correspondence to Ilya Kachkovskiy.

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The first author was supported by the RSF 22-11-00092 grant.

The second author was partially supported by the National Science Foundation DMS–1846114 and DMS–2052519 Grants, and the Sloan Research Fellowship 2022. The second author’s research visits to St. Petersburg were partially supported by the RSF 17-11-01069 Grant and the program “Spectral Theory and Mathematical Physics” at Euler Institute, St. Petersburg.

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Filonov, N., Kachkovskiy, I. On Spectral Bands of Discrete Periodic Operators. Commun. Math. Phys. 405, 21 (2024). https://doi.org/10.1007/s00220-023-04891-7

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