Semiclassical spectral estimates for Toeplitz operators

Borthwick, David; Paul, Thierry; Uribe, Alejandro

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

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 ${\mathscr{H}}_k=H^0(X,L^{\otimes k})$. 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.

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 ${\mathscr{H}}_k=H^0(X,L^{\otimes k})$. 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.

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