论文标题
本征限制限制的傅立叶系数
Fourier coefficients of restrictions of eigenfunctions
论文作者
论文摘要
令$ \ {e_j \} $为紧凑型riemannian歧管$（m，g）$的拉普拉斯特征函数的正常基础。令$ h \ subset m $为子手机，让$ \ {ψ_k\} $成为诱导度量的$ h $的laplace eigenfunctions的正常基础。我们获得了傅里叶系数的联合渐近学\ [ \langleγ_He_j，ψ_k\ rangle_ {l^2（h）} = \ int_h e_j \ overlineψ_k\，dv_h，dv_h，\ \]，$γ_he_j $ of $ e_j $ of $ e_j $ of $ e_j $ of $ e_j $ of $ e_j $ to $ e_j $ to $ h $。特别是，我们获得了关节频谱$ \ {（μ_k，λ_j）\} _ {J，k -0}^{\ nifty $ $ $ minu $ um min $ min $ minth $ minth $ minth $ min和y的$ min和up minth $ min和y的$ min和y $ minth $ min和y $ minth $ min和the $ minth $ min和y $ minth $ min和y $ minth $ min和y $ minth $ min和y $ minth $ min和y $ m in $ min和y $ minth $ min和y $ min和y $ m in $ min和raph $ min和y $ m in $ min和y $ min和uph， $ \ Mathbb r^2 $的一个合适的“厚”区域。较厚的区域包括（1）[a，b] \ subset（0,1）$和$λ_j\leqλ$，以及（2）缓慢增稠的条$ |μ_k-cλ_j| \ leq w（λ）$和$λ_j\leqλ$，其中$ w（λ）$是单调的，$ 1 \ ll w（λ）\lyseSimλimλ^{1-1-1/n} $。获取这些渐近学的关键工具包括傅立叶积分算子的组成和新的多维陶拜定理。
Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $Δ_M$ on $M$ and the Laplacian $Δ_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $μ_k/λ_j \in [a,b] \subset (0,1)$ and $λ_j \leq λ$, and (2) the slowly thickening strip $|μ_k - cλ_j| \leq w(λ)$ and $λ_j \leq λ$, where $w(λ)$ is monotonic and $1 \ll w(λ) \lesssim λ^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.
