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Wei-Chuan Shen

Publications and source records attributed to Wei-Chuan Shen.

5 recordsLinked to original sources
arXiv2025

Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds

We consider any compact CR manifold whose Levi form is non-degenerate of constant signature $(n_-,n_+)$, $n_-+n_+=n$. For $λ>0$ and $q\in\{0,\cdots,n\}$, we let $Π_λ^{(q)}$ be the spectral projection of the Kohn Laplacian of $(0,q)$-forms corresponding to the interval $[0,λ]$. For certain classical pseudodifferential operators $P$, we study a class of generalized elliptic Toeplitz operators $T_{P,λ}^{(q)}:=Π_λ^{(q)}\circ P\circ Π_λ^{(q)}$. For any cut-off $χ\in\mathscr C^\infty_c(\mathbb R\setminus\{0\})$, we establish the full asymptotics of the semi-classical spectral projector $χ(k^{-1}T_{P,λ}^{(q)})$ as $k\to+\infty$. Our main result conclude that the smooth Schwartz kernel $χ(k^{-1}T_{P,λ}^{(n_-)})(x,y)$ is the sum of two semi-classical oscillatory integrals with complex-valued phase functions.

math.CV↗
arXiv2024

Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions

Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let $T_P$ be the Toeplitz operator on $X$ associated with a first-order pseudodifferential operator $P$. In our previous work we established the asymptotic expansion for $k$ large of the kernel of the operators $χ(k^{-1}T_P)$, where $χ$ is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that $X$ can be projectively embedded by maps with components of the form $χ(k^{-1}λ)f_λ$, where $λ$ is an eigenvalue of $T_P$ and $f_λ$ is a corresponding eigenfunction. We establish the asymptotics of the pull-back of the Fubini-Study metric by these maps and we obtain the distribution of the zero divisors of random Cauchy-Riemann functions. We then establish a version of the Lelong-Poincaré formula for domains with boundary and obtain the distribution of the zero divisors of random holomorphic functions on strictly pseudoconvex domains.

math.CV↗
arXiv2023

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 $χ_k(T_P)$ 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 $k\to+\infty$. 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.

math.CV↗
arXiv2019

Asymptotics of torus equivariant Szegő kernel on a compact CR manifold

For a compact CR manifold $(X,T^{1,0}X)$ of dimension $2n+1$, $n\geq 2$, admitting a $S^1\times T^d$ action, if the lattice point $(-p_1,\cdots,-p_d)\in\mathbb{Z}^{d}$ is a regular value of the associate CR moment map $μ$, then we establish the asymptotic expansion of the torus equivariant Szegő kernel $Π^{(0)}_{m,mp_1,\cdots,mp_d}(x,y)$ as $m\to +\infty$ under certain assumptions of the positivity of Levi form and the torus action on $Y:=μ^{-1}(-p_1,\cdots,-p_d)$.

math.CV↗