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Low-energy spectrum of Toeplitz operators

Abstract : Berezin-Toeplitz operators allow to quantize functions, or symbols, on compact Kähler manifolds, and are defined using the Bergman (or Szeg\H{o}) kernel. We study the spectrum of Toeplitz operators in an asymptotic regime which corresponds to a semiclassical limit. This study is motivated by the atypic magnetic behaviour observed in certain crystals at low temperature. We study the concentration of eigenfunctions of Toeplitz operators in cases where subprincipal effects (of same order as the semiclassical parameter) discriminate between different classical configurations, an effect known in physics as quantum selection . We show a general criterion for quantum selection and we give detailed eigenfunction expansions in the Morse and Morse-Bott case, as well as in a degenerate case. We also develop a new framework in order to treat Bergman kernels and Toeplitz operators with real-analytic regularity. We prove that the Bergman kernel admits an expansion with exponentially small error on real-analytic manifolds. We also obtain exponential accuracy in compositions and spectra of operators with analytic symbols, as well as exponential decay of eigenfunctions.
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Submitted on : Tuesday, October 15, 2019 - 10:22:07 AM
Last modification on : Wednesday, July 29, 2020 - 2:49:45 PM
Long-term archiving on: : Friday, January 17, 2020 - 11:41:03 AM


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  • HAL Id : tel-02053539, version 2



Alix Deleporte-Dumont. Low-energy spectrum of Toeplitz operators. Spectral Theory [math.SP]. Université de Strasbourg, 2019. English. ⟨NNT : 2019STRAD004⟩. ⟨tel-02053539v2⟩



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