Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains

2023/08/18 by Hsiao, Chin-Yu, Marinescu, George · 1 citation
#Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.09820

Abstract

On a relatively compact strictly pseudoconvex domain with smooth boundary in a complex manifold of dimension n we consider a Toeplitz operator T_R with symbol a Reeb-like vector field R near the boundary. We show that the kernel of a weighted spectral projection χ(k^-1T_R), where χ is a cut-off function with compact support in the positive real line, is a semi-classical Fourier integral operator with complex phase, hence admits a full asymptotic expansion as k→+∞. More precisely, the restriction to the diagonal χ(k^-1T_R)(x,x) decays at the rate O(k^-∞) in the interior and has an asymptotic expansion on the boundary with leading term of order kⁿ⁺¹ expressed in terms of the Levi form and the pairing of the contact form with the vector field R.

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