Eigenfunction Asymptotics and Quantum Chaos

从广义上讲，我的研究主要集中在半经典极限下薛定谔算子的特征函数的研究。我主要对渐近增长和积累性质以及节点集和临界集的行为感兴趣。我建议继续我在这个领域的工作，更具体地说，我计划关注两个不同（但相关）的研究方向：1)特征函数限制界：设$(M,g)$是一个紧流形，具有拉普拉斯-贝尔特拉米算子$-\Delta$和$\phi_{\lambda}$是一个$L^2$规范化的特征函数，特征值$\lambda^2$和$H \子集M$是一个光滑超曲面。从上到下估计$L^2$-限制界$ \| \phi_{\lambda} \|_{L^2(H)}$的问题在特征函数节点（即零）和临界集[TZ1， ET， JZ， TZ2]的研究中有许多重要的应用。$\ | \phi_{\lambda} \|_{L^2(H)} \geq e^{-C \lambda}$对于所有$\phi_{\lambda}$与$\lambda \geq \lambda_0$的本征函数限制的“良度”界是建立与超曲面H的节点相交上界的一个非常重要的组成部分。最近，在与S. Zelditch [TZ2]的合作中，我们在这个问题上取得了重要进展。但是，这种估计是否在一般情况下得到普遍满足的问题仍然没有解决。我打算在$\ \上的狄利克雷特征函数的情况下研究这个问题，在超曲面$H$接近边界$\偏$的情况下。(b)上界：$\| \phi_{\lambda} \|_{L^2(H)|$的通称上界[BGT]的改进也是本征函数振荡研究的核心。特别是，获得沿边界的超曲面的改进问题是特别重要的。最近在[CT]中，我们在$H \子集\偏\ ω $是完全测地线和$\ ω $是分段光滑凸平面域的情况下建立了改进。我建议将这些结果推广到更一般的有边界流形。2)内部STEKLOV特征函数的节点结构设$\ ω $是一个紧致光滑流形，其边界$\偏\ ω = m $最近，有大量研究与Dirichlet-to-Neumann （DtN）或STEKLOV算子的谱渐近性以及相应的特征函数节点集有关的活动。在与Polterovich和Sher [PST]的联合工作中，我们最近证明了在$\ ω $是具有实解析边界的黎曼曲面的情况下，对于内部Steklov特征函数的节点集的Yau猜想的尖锐模拟。在高维中，对于下界的情况我们所知甚少。我建议利用Galkowski [GT]关于Steklov特征函数的尖锐点界的最新结果来研究这个问题。

Summary of Proposal (required) Broadly speaking, my research is focused on the study of eigenfunctions of Schrodinger operators in the semiclassical limit. I am primarily interested in the asymptotic growth and accumulation properties as well as the behaviour of the nodal and critical sets. I propose to continue my work in this field and more specifically, I plan to focus on two different (but related) lines of research: 1) EIGENFUNCTION RESTRICTION BOUNDS: Let $(M,g)$ be a compact manifold with Laplace-Beltrami operator $-\Delta$ and $\phi_{\lambda}$ be an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ and $H \subset M$ a smooth hypersurface. The problem of estimating $L^2$-restriction bounds $ \| \phi_{\lambda} \|_{L^2(H)}$ from above and below has many important applications in the study of eigenfunction nodal (i.e. zero) and critical sets [TZ1, ET, JZ, TZ2] (a) Lower bounds: Proving unique continuation (i.e. ``goodness" bounds) for eigenfunction restrictions of the form $ \| \phi_{\lambda} \|_{L^2(H)} \geq e^{-C \lambda}$ for all $\phi_{\lambda}$ with $\lambda \geq \lambda_0$ is a very important ingredient in establishing upper bounds for nodal intersections with the hypersurface H. Recently, in joint work with S. Zelditch [TZ2], we have made important progress on this problem. However, the question of whether such estimates are generically satisfied in a general setting remains open. I propose to investigate this in the case of Dirichlet eigenfunctions on a domain $\Omega$ in the case of hypersurfaces $H$ close to the boundary $\partial \Omega. (b) Upper bounds: Improvements in universal upper bounds [BGT] for $\| \phi_{\lambda} \|_{L^2(H)|$ are also central to the study of eigenfunction oscillations. In particular, the question of obtaining improvements for hypersurfaces along the boundary is of particular importance. Recently in [CT], we have established improvements in the case where $H \subset \partial \Omega$ is totally-geodesic and $\Omega$ is a piecewise-smooth convex planar domain. I propose to extend these results to more general manifolds with boundary. 2) NODAL STRUCTURE OF INTERIOR STEKLOV EIGENFUNCTIONS Let $\Omega$ be a compact, smooth manifold with boundary $\partial \Omega = M.$ Recently, there has a great deal of activity related to the spectral asymptotics of the associated Dirichlet-to-Neumann (DtN)  or Steklov operator and the study of corresponding eigenfunction nodal sets.  In joint work with Polterovich and Sher [PST], we have recently proved the sharp analogue of the Yau conjecture for nodal sets for interior Steklov eigenfunctions in the case when $\Omega$ is a Riemann surface with real-anaytic boundary. In higher dimensions, very little  is known in the case of lower bounds. I propose to investigate this question using recent results with Galkowski [GT] on sharp pointwise bounds for Steklov eigenfunctions.