On the second coefficient in the semi-classical expansion of toeplitz operators

• Published in: Analysis and mathematical physics Vol. 15; no. 4
• Main Authors: Chang, Chin-Chia, Herrmann, Hendrik, Hsiao, Chin-Yu
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.08.2025
• Subjects: Asymptotic series; Fields (mathematics); Operators (mathematics)
• ISSN: 1664-2368; 1664-235X
• DOI: 10.1007/s13324-025-01105-2

Let X be a compact strictly pseudoconvex embeddable CR manifold and let A be the Toeplitz operator on X associated with a Reeb vector field $${\mathcal {T}}\in {\mathscr {C}}^\infty (X,TX)$$ T ∈ C ∞ ( X , T X ) . Consider the operator $$\chi _k(A)$$ χ k ( A ) defined by the functional calculus of A , where $$\chi $$ χ is a smooth function with compact support in the positive real line and $$\chi _k(\lambda ):=\chi (k^{-1}\lambda )$$ χ k ( λ ) : = χ ( k - 1 λ ) . It was established recently that $$\chi _k(A)(x,y)$$ χ k ( A ) ( x , y ) admits a full asymptotic expansion in k when $$k$$ k becomes large. The second coefficient of the expansion plays an important role in the further studies of CR geometry. In this work, we calculate the second coefficient of the expansion.