Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

2023/03/30 by Herrmann, Hendrik, Hsiao, Chin-Yu, Marinescu, George +1 · 1 citation
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2303.17319

Abstract

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 χ_k(T_P) the functional calculus of T_P by any rescaled cut-off function χ with compact support in the positive real line. In this work, we show that χ_k(T_P) admits a full asymptotic expansion as k→+∞. 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.

Cited by

• Bernstein-Markov measures and Toeplitz theory

Related

• On the second coefficient in the semi-classical expansion of Toeplitz Operators
• On the distribution kernels of Toeplitz operators on CR manifolds
• Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds
• Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains
• Szegö kernel asymptotics for high power of CR line bundles and Kodaira embedding theorems on CR manifolds