论文标题
szegő本征素的内核渐近学和浓度
Szegő kernel asymptotics and concentration of Husimi Distributions of eigenfunctions
论文作者
论文摘要
我们在一个封闭的，真实的分析riemannian歧管$ m $的grauert管的边界$ \ partialm_τ$上工作。与reeb vector字段相关的toeplitz运算符$π_τd_ {\sqrtρ}π_τ$是$ h^2（\ partialm_τ）$的积极的，自动化的，椭圆的操作员。我们在抛物线（Reeb）流量（REEB）流中的抛物面重新分组$ g^{t}_τ= \ exp tT面_ {\sqrtρ} $中，$λ\ to \ infty $渐近ry渐差重新分组。 π_τ$。我们还计算了在$ m $ $ \ laplace eigenfunctions上的$ \ partialm_τ$上的husimi分布的钢化和缩放级别渐变学。这两个渐近公式都可以用Bargmann- fock空间上的地球流量$ g^{t}_τ$的线性化表示。作为推论，我们获得了$π_{χ，λ} $的尖锐$ l^p \ to l^{q} $ norm估算，而Husimi发行版的夏普$ l^p $估计值。
We work on the boundary $\partial M_τ$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_τ)$. We compute $λ\to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_τ = \exp tΞ_{\sqrtρ}$ for the spectral projection kernel $Π_{χ, λ}$ associated to $Π_τD_{\sqrtρ} Π_τ$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_τ$ of Laplace eigenfunctions on $M$. Both asymptotic formulae can be expressed in terms of the metaplectic representation of the linearization of the geodesic flow $G^{t}_τ$ on Bargmann--Fock space. As a corollary, we obtain sharp $L^p \to L^{q}$ norm estimates for $Π_{χ, λ}$ and sharp $L^p$ estimates for Husimi distributions.
