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Complex Analysis and Geometry Seminar

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Professor Chin-Yu Hsiao

Location:  Hill 705
Date & time: Tuesday, 29 August 2023 at 2:00PM - 3:00PM

Abstarct: Let \(X\) be a compact strictly pseudoconvex embeddable CR manifold and let

\(T_P\) be the Toeplitz operator on \(X\) associated with a first order pseudodifferential operator \(P\). We consider the operator \(\chi_k(T_P)\) defined by functional calculus of \(T_P\), where \(\chi\) is a smooth function with compact support in the positive real line and \(\chi_k(\lambda):=\chi(k^{-1}\lambda)\). We

show that \(\chi_k(T_P)\) admits a full asymptotic expansion as \(k\to+\infty\). As applications, we obtain several CR analogues of results concerning the high powers of line bundles in complex geometry. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and an embedding theorem of strictly pseudoconvex CR manifolds into perturbed spheres. 

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