Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Let \(X\) be a compact strictly pseudoconvex embeddable CR manifold and let \(T_P\) be the Toeplitz operator on \(X\) associated with some first order pseudodifferential operator \(P\). We consider \(\chi_k(T_P)\) the functional calculus of \(T_P\) by any rescaled cut-off function \(\chi\) with comp...

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Bibliographic Details
Published inarXiv.org
Main Authors Herrmann, Hendrik, Chin-Yu, Hsiao, Marinescu, George, Wei-Chuan Shen
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.12.2023
Subjects
Asymptotic series
Embedding
Operators (mathematics)
Theorems
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Summary:Let \(X\) be a compact strictly pseudoconvex embeddable CR manifold and let \(T_P\) be the Toeplitz operator on \(X\) associated with some first order pseudodifferential operator \(P\). We consider \(\chi_k(T_P)\) the functional calculus of \(T_P\) by any rescaled cut-off function \(\chi\) with compact support in the positive real line. In this work, we show that \(\chi_k(T_P)\) admits a full asymptotic expansion as \(k\to+\infty\). As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.
ISSN:2331-8422