Spectral asymptotics of semiclassical unitary operators

Abstract : We introduce axiomatically a semiclassical quantization for non necessarily self-adjoint operators. Then we focus on unitary operators and prove that, in the semiclassical limit, the convex hull of the joint spectrum of a finite commuting family of semiclassical unitary operators converges to the convex hull of the joint image of the principal symbols, which can be shown to be a subset of a d-torus $(\mathbb{S}^1)^d$. This result covers in particular $\hbar$-pseudodifferential and Berezin-Toeplitz operators. Part of the paper is devoted to the definition of this notion of convex hull for subsets of tori. The proof of our result builds on recent results for semiclassical self-adjoint operators and involves the inverse Cayley transform for unitary operators.
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https://hal.archives-ouvertes.fr/hal-01161621
Contributeur : Yohann Le Floch <>
Soumis le : vendredi 19 juin 2015 - 21:54:59
Dernière modification le : mardi 10 avril 2018 - 01:09:17
Document(s) archivé(s) le : mardi 25 avril 2017 - 18:00:49

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LFPelayo2015_final_v2.pdf
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  • HAL Id : hal-01161621, version 2
  • ARXIV : 1506.02873

Citation

Yohann Le Floch, Alvaro Pelayo. Spectral asymptotics of semiclassical unitary operators. 43 pages, 5 figures. 2015. 〈hal-01161621v2〉

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