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Low-Energy Spectrum of Toeplitz Operators with a Miniwell

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Abstract

We study the concentration properties of low-energy states for quantum systems in the semiclassical limit, in the setting of Toeplitz operators, which include quantum spin systems as a large class of examples. We establish tools proper to Toeplitz quantization to give a general subprincipal criterion for localisation. In addition, we build up symplectic normal forms in two particular settings, including a generalisation of Helffer–Sjöstrand miniwells, in order to prove asymptotics for the ground state and estimates on the number of low-lying eigenvalues.

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Acknowledgements

The authors thanks his thesis committee and B. Helffer for fruitful discussion.

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Authors and Affiliations

  1. CNRS, IRMA UMR 7501, Université de Strasbourg, 67000, Strasbourg, France

    Alix Deleporte

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  1. Alix Deleporte
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Correspondence to Alix Deleporte.

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Communicated by J. Marklof

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This work was supported by Grant ANR-13-BS01-0007-01.

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Deleporte, A. Low-Energy Spectrum of Toeplitz Operators with a Miniwell. Commun. Math. Phys. 378, 1587–1647 (2020). https://doi.org/10.1007/s00220-020-03791-4

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