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Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators

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From Complex Analysis to Operator Theory: A Panorama

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 291))

Abstract

In this survey, we give an introduction to and proof of the gap labelling theorem for discrete one-dimensional ergodic Schrödinger operators via the Schwartzman homomorphism. To keep the paper relatively self-contained, we include background on the integrated density of states, the oscillation theorem for 1D operators, and the construction of the Schwartzman homomorphism. We illustrate the result with some examples. In particular, we show how to use the Schwartzman formalism to recover the classical gap-labelling theorem for almost-periodic potentials. We also consider operators generated by subshifts and operators generated by affine homeomorphisms of finite-dimensional tori. In the latter case, one can use the gap-labelling theorem to show that the spectrum associated with potentials generated by suitable transformations (such as Arnold’s cat map) is an interval.

Dedicated to the memory of Sergey Naboko

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Notes

  1. 1.

    We employ a minor but standard abuse of notation, writing \(\alpha \) both for the element of \({\mathbb R}^d\) and its projection to \({\mathbb T}^d\). This will be convenient later on when we talk about flows.

  2. 2.

    To be more precise, while \({\mathfrak {A}}\) does not have an immediate analogue in higher dimensions in general, in the specific setting considered in these papers, \({\mathfrak {A}}\) can be identified as a set, compare (1.12), that does have a natural higher-dimensional counterpart.

  3. 3.

    The trace is at most two since the solution space of \(H_\omega \psi = E_0\psi \) is two-dimensional; this bound already suffices for this argument. One can reduce the two to one by using constancy of the Wronskian.

  4. 4.

    Recall that we write \(\alpha \) both for the vector in \({\mathbb R}^d\) and its projection to \({\mathbb T}^d\).

  5. 5.

    This is well known and not hard to show using that any map \({\mathbb T} \to {\mathbb T}\) is homotopic to \(x\mapsto nx\) for some \(n \in {\mathbb Z}\). Alternatively, it also follows from the more general discussion below in Sect. 6.3.

  6. 6.

    We are writing the abelian group \(\Omega \) additively, so one should understand the zero in the first coordinate as the additive identity element of \(\Omega \).

  7. 7.

    Notice that this is stronger than just proving uniform continuity of \(\widetilde G\), since \(d([\alpha ,t],[\alpha ,s])\) can be small even if \(s-t\) is not small.

  8. 8.

    This uses \(A0=0\).

  9. 9.

    Indeed, the spectrum of any Schrödinger operator in \(\ell ^2({\mathbb Z})\) with a non-constant periodic potential has at least one gap [43].

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Acknowledgements

We are grateful to Michael Baake, Franz Gähler, Svetlana Jitomirskaya, Johannes Kellendonk, Andy Putman, Lorenzo Sadun, and Hiro Lee Tanaka for helpful conversations and comments. We also want to thank the American Institute of Mathematics for hospitality and support during a January 2022 visit, during which part of this work was completed. We also gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University at which some of this work was performed.

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Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, USA

    David Damanik

  2. Department of Mathematics, Texas State University, San Marcos, TX, USA

    Jake Fillman

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Editors and Affiliations

  1. School of Computer Science & Informatics, Cardiff University, Cardiff, UK

    Malcolm Brown

  2. Department of Mathematics, Baylor University, Waco, TX, USA

    Fritz Gesztesy

  3. Department of Mathematics, Stockholm University, Stockholm, Sweden

    Pavel Kurasov

  4. Department of Mathematics Imperial College London London, United Kingdom, Sirius Mathematical Center, Sochi, Russia

    Ari Laptev

  5. Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, USA

    Barry Simon

  6. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, USA

    Gunter Stolz

  7. School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, UK

    Ian Wood

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Damanik, D., Fillman, J. (2023). Gap Labelling for Discrete One-Dimensional Ergodic Schrödinger Operators. In: Brown, M., et al. From Complex Analysis to Operator Theory: A Panorama. Operator Theory: Advances and Applications, vol 291. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-31139-0_14

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