Semiclassical spectral estimates for Toeplitz operators

Borthwick, David; Paul, Thierry; Uribe, Alejandro

doi:10.5802/aif.1654

Borthwick, David ; Paul, Thierry ; Uribe, Alejandro

Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 1189-1229.

Résumé
Abstract

Soit $X$ une variété kählérienne compacte de classe de Kähler entière et $L \to X$ un fibré en droites hermitien holomorphe, dont la courbure est la forme symplectique sur $X$ . Soit $H \in C^{\infty} (X, ℝ)$ un hamiltonien et $T_{k}$ l’opérateur de Toeplitz de multiplicateur $H$ agissant sur l’espace $ℋ_{k} = H^{0} (X, L^{\otimes k})$ . On obtient des estimations sur les valeurs et fonctions propres de $T_{k}$ lorsque $k \to \infty$ en termes du flot hamiltonien associé a $H$ . On étudie en détail le cas où $X$ est une orbite coadjointe entière d’un groupe de Lie.

Let $X$ be a compact Kähler manifold with integral Kähler class and $L \to X$ a holomorphic Hermitian line bundle whose curvature is the symplectic form of $X$ . Let $H \in C^{\infty} (X, ℝ)$ be a Hamiltonian, and let $T_{k}$ be the Toeplitz operator with multiplier $H$ acting on the space $ℋ_{k} = H^{0} (X, L^{\otimes k})$ . We obtain estimates on the eigenvalues and eigensections of $T_{k}$ as $k \to \infty$ , in terms of the classical Hamilton flow of $H$ . We study in some detail the case when $X$ is an integral coadjoint orbit of a Lie group.

EuDML MR Zbl

DOI : 10.5802/aif.1654

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     author = {Borthwick, David and Paul, Thierry and Uribe, Alejandro},
     title = {Semiclassical spectral estimates for {Toeplitz} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {1189--1229},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {4},
     year = {1998},
     doi = {10.5802/aif.1654},
     zbl = {0920.58059},
     mrnumber = {2000c:58048},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1654/}
}

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Borthwick, David; Paul, Thierry; Uribe, Alejandro. Semiclassical spectral estimates for Toeplitz operators. Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 1189-1229. doi : 10.5802/aif.1654. https://aif.centre-mersenne.org/articles/10.5802/aif.1654/

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