环面本征函数对超曲面的限制

Averages of eigenfunctions over hypersurfaces

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{ ϕ_h \}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $μ$. Let $H$ be a smooth hypersurface. Our main result says that when $μ$ is $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.

解析流形上本征函数节点集的分布

Distribution of the nodal sets of eigenfunctions on analytic manifolds

紧黎曼流形上的Laplacian本征函数的节集具有余维数1和有限超曲面测度。本文研究了当流形上的度规是解析的时，特征函数节点集的分布。我们证明了，如果本征函数在小尺度上是等分布的，那么它们的节点集的超曲面体积形式的弱极限与流形上的体积形式是可比的。特别地，在具有解析度量的负弯曲流形和环面上，我们证明了在任何特征基中，存在特征函数的全密度子序列，使得其节点集的超曲面体积形式的弱极限与流形上的体积形式相当。

超曲面上Laplace算子的极限吸收原理、广义本征函数和散射矩阵

Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

本文给出了紧致超曲面$Γ=\partialΩ$，$Ω\subset{\mathbb{R}}^{n}$上Laplace算子的自伴实现的极限吸收原理.对于任何这样的自伴算子，我们还提供了广义本征函数和散射矩阵，这两个对象都写在算子值外尔函数。我们利用一个Krein型公式，它提供了对应于超曲面上的自伴边界条件的算子与整个空间${\mathbb{R}}^{n}$上的自由拉普拉斯算子之间的预解式差。我们的结果适用于所有标准的边界条件的例子，如狄利克雷，诺依曼，罗宾，$δ$和$δ'$型，无论是分配在$Γ$或$Σ\subsetΓ$。

超曲面上Neumann数据的外质量估计和$L^2$限制界

Exterior mass estimates and $L^2$ restriction bounds for Neumann data along hypersurfaces

研究了紧黎曼流形$M$上Laplace本征函数的$L^2$范数的估计问题。我们证明了质量估计的限制本征函数$ϕ_h$，$(h^2 Δ- 1)ϕ_h = 0$，$H$在该地区以外的COBALL丛的$H$，在$h^δ$尺度（$0\leq δ< 2/3$）。我们使用这个估计得到一个$O(1)$ $L^2$ -限制界的诺依曼数据沿$H.$的估计也适用于半经典薛定谔算子的本征函数。

Planck-scale distribution of nodal length of arithmetic random waves

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use o…

Lower bounds for Steklov eigenfunctions

Let $(Ω,g)$ be a compact, analytic Riemannian manifold with analytic boundary $\partial Ω= M.$ We give $L^2$-lower bounds for Steklov eigenfunctions and their restrictions to interior hypersurfaces $H \subset Ω^{\circ}$ in a geometrically defined neighborhood of $M$. Our results are optimal in the entire geometric neighborhood and complement the results on eigenfunction upper bounds in the author's previous work.

超曲面收缩管状邻域中的磁Laplacian

The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces

在有磁场的任意维欧氏空间中，当两个平行超曲面之间的距离趋于零时，讨论了它们之间的Dirichlet Laplacian的极限.我们表明，拉普拉斯算子收敛在一个规范的预解式意义上的薛定谔算子的限制超曲面的电磁势表示的主曲率和投影的环境矢量潜在的超曲面。作为应用，我们得到了薄量子层中束缚态能量和本征函数的有效近似。

量子遍历拉普拉斯特征函数解析延拓的$L^2$限制界

$L^2$ restriction bounds for analytic continuations of quantum ergodic Laplace eigenfunctions

本文证明了实超曲面上的量子遍历限制定理，并应用于获得拉普拉斯特征函数FBI变换在满足一般几何条件超曲面上的$L^2$限制的$h$无关上下界。

特征函数在扫视集附近的$L^2$行为

The $L^2$ Behavior of Eigenfunctions Near the Glancing Set

设$M$是有边界或无边界的紧致流形，$H\subset M$是光滑的内超曲面.研究了拉普拉斯本征函数解$(-h^2Δ_g-1)u=0$到$H$的限制。特别是，我们研究退化的$u|_H$作为一个microlocalized方法，通过寻找最佳的电源$s_0$，使$(1+h^2Δ_H)_+^{s_0}u|_H$保持一致有界的$L^2(H)$作为$h\to 0$的掠集。此外，我们表明，这个界限是饱和的，在每一个$h$依赖的规模附近掠用的例子上的磁盘和球。我们给我们的估计量子遍历限制定理的应用。

薄曲层节点集

Nodal sets of thin curved layers

本文讨论了任意维欧几里得空间中Dirichlet Laplacian的特征函数在超曲面的细管状邻域中的节点集的位置。在邻域半径趋于零的极限中，已知拉普拉斯算子的谱性质可以通过超曲面上的有效薛定谔算子很好地近似，该超曲面具有仅用主曲率表示的势。利用椭圆型偏微分方程的技巧，加强了已有的扰动结果，得到了特征函数在Hölder空间中的收敛性.这使我们特别能够得出结论，每个节点集与管状邻域的边界有一个非空的交集。

跟踪特征值到模空间的边界I：收敛性和谱积累

Tracking eigenvalues to the frontier of moduli space I: Convergence and spectral accumulation

研究了超曲面上退化的黎曼度量族的拉普拉斯算子的本征函数/本征值的极限行为。我们的结果推广了早期关于双曲曲面退化的工作。

High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology

Let $Σ$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $Σ$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $ψ$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $Δψ=-λψ$ (with $λ=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $ψ$ given by~$Σ$, up to an ambient diffeomorphism.

Reverse Agmon estimates and nodal intersection bounds in forbidden regions

Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \mathbb{R})$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 Δ_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in th…

Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty}$ bounded planar domains

Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for…

负曲率曲面上 Neumann 数据的不变节线不存在性与改进的 L^2 限制下界

Nonexistence of invariant nodal line and improved $L^2$ restriction bounds for Neumann data on negatively curved surface

本文针对负曲率曲面上的拉普拉斯特征函数限制问题，证明了不变节线不存在，并获得了 Neumann 数据 L^2 限制下界的定量改进。