Berezin-Toeplitz quantization for lower energy forms

Chin-Yu Hsiao InInstitute of Mathematics, Academia Sinica, Taiwan George Marinescu University of Cologne, Germany

Complex Variables and Complex Analysis mathscidoc:1803.08005

Comm. Partial Differential Equations , 42, (6), 895-942, 2017
Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$--forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg\H{o} kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\H{o} kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\Complex^N$, for some $N\in\mathbb N$.
Geometric quantization theory, Semi-classical Analysis, Fourier integral operators
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  title={Berezin-Toeplitz quantization for lower energy forms},
  author={Chin-Yu Hsiao, and George Marinescu},
  booktitle={ Comm. Partial Differential Equations },
Chin-Yu Hsiao, and George Marinescu. Berezin-Toeplitz quantization for lower energy forms. 2017. Vol. 42. In Comm. Partial Differential Equations . pp.895-942.
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