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Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

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Abstract

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

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Acknowledgements

The authors thank Franz Luef, Domenico Monaco and Guo Chuan Thiang for valuable feedback on an earlier version of this manuscript. We also thank Giovanna Marcelli, Massimo Moscolari and Gianluca Panati for sharing the results of [29, 30] with us. CB is supported by a JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14548) and thanks the Mathematical Institute, Universiteit Leiden, for hospitality during the conference Noncommutative Geometry, Analysis, and Topological Insulators in February 2020, where this work took shape. Both authors thank the Casa Matematica Oaxaca for hospitality and support during the workshop Topological Phases of Interacting Quantum Systems in June 2019.

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Authors and Affiliations

  1. WPI-AIMR, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

    Chris Bourne

  2. RIKEN iTHEMS, 2-1 Hirosawa, Wako, Saitama, 351-0198, Japan

    Chris Bourne

  3. Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands

    Bram Mesland

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  1. Chris Bourne
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Correspondence to Chris Bourne.

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Bourne, C., Mesland, B. Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences. J Fourier Anal Appl 27, 69 (2021). https://doi.org/10.1007/s00041-021-09873-8

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