Eigenfunction Asymptotics and Quantum Chaos

概括地说，我的研究主要集中在研究薛定谔算子在半经典极限下的本征函数。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。我打算继续我在这个领域的工作，更具体地说，我计划集中在两个不同的(但相关的)研究方向：1)特征函数限制界：设$(M，g)$是具有Laplace-Beltrami算子的紧致流形$-\Delta$，$是$L^2$-正规化的本征函数，其特征值$\lambda^2$和$H\子集M$是光滑超曲面。上下界的估计问题在特征函数结点(即零)和临界集[TZ1，Et，JZ，[TZ1，Et，JZ]，[TZ1，Et，JZ，]中有着重要的应用(A)下界：证明了具有超曲面H的特征函数约束的唯一延拓(即“良界”)。最近，在与S.Zelditch[TZ2]的合作中，我们在这个问题上取得了重要的进展。然而，这样的估计是否在一般情况下得到普遍满足的问题仍然悬而未决。我建议在区域$\Omega$上的Dirichlet特征函数的情况下，在靠近边界$\Partial\Omega的超曲面$H$的情况下研究这一点。(B)上界：对L^2(H)的泛函上界[BGT]的改进也是研究本征函数振荡的核心。特别是，获得边界沿线超曲面的改进问题特别重要。最近在[CT]中，我们在$H\SUBSET\PARTIAL\Omega$是全测地且$\Omega$是分片光滑凸平面区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带边界的流形上。2)内部Steklov特征函数的结点结构设Omega是一个紧致光滑流形，边界为$\Partial\Omega=M。最近，关于伴随的Dirichlet-to-Neumann(DTN)算子或Steklov算子的谱渐近性态以及相应的特征函数结点集的研究，有许多活跃的活动。最近，我们与Polterovich和Sher[PST]合作，证明了内部Steklov特征函数的节点集的Yau猜想在欧米茄曲面为实-解析边界的情形下的类似结果。在更高的维度中，对于下限的情况，我们知道的很少。我建议使用Galkowski[GT]关于Steklov特征函数的逐点锐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.