Eigenfunction asymptotics and quantum chaos

*我的基本研究兴趣涉及模拟电子密度概率分布的量子波函数的高能行为。我最感兴趣的问题是如何在高能极限下估计波函数的幅度和频率。更准确地说，令$(M，g)$是紧致的，具有Laplace-Beltrami算子的黎曼流形$\Delta_g。$我提议的研究计划集中于相关Laplace本征函数的渐近和相应本征值的Weyl渐近。我的研究建议的很大一部分集中在本征函数的渐近限制界(包括上下界)。在过去的几年里，本征函数限制界已经成为量子混沌中一个非常活跃的领域，不仅是因为它们本身的兴趣，而且因为它们在本征函数节点域和临界集的渐近性方面的广泛应用。在最近与Zelditch和El-Hajj的工作中，I i证明了沿着实解析曲线(即.“优度估计”)直接控制特征函数节点集与曲线的交点个数。通过明智地选择曲线，例如在算术曲面的情况下，Ghosh，Reznikov和Sarnak最近证明了一个人可以用这样的交点对来反转相关的节点域。这导致了该领域一个非常激动人心的新发展：对库朗特节点定理的部分颠倒。因此，我建议研究核心问题：*问题1：给定黎曼曲面$(M，g)$上的一条曲线$H$，在什么条件下，$H$一定是一条好的曲线？*问题2：在什么条件下，人们可以在库朗特定理中建立一个定量下界？*有一些特别令人感兴趣的情况。最近，我们用El-Hajj肯定地回答了问题1，其中$H$是严格凸的，且环境流形是具有遍历Bliiard动力学的分段光滑平面域。然而，其他情况仍然完全开放。我建议在本征函数序列是量子遍历的其他情况下同时调查问题1和2。最后，由于谱的多重性和选择本征函数基的自由度，一般球谐函数的情况应该被证明是特别吸引人的和丰富的试验场。和我的几个学生一起，我建议在后一种情况下也调查这两个问题。

***My basic research interests involve the high-energy behaviour of quantum wavefunctions that model the probabilistic distribution of electron density.  I am most interested in the problem  of  estimating  wavefunction amplitudes and frequencies in the high-energy limit.*** *** More precisely, let $(M,g)$ be compact, Riemannian manifold with Laplace-Beltrami operator $\Delta_g.$ My proposed research program is focused on the asymptotics of the associated Laplace  eigenfunctions and Weyl asymptotics of the corresponding eigenvalues.  A substantial part of my research proposal is focused on asymptotic  restriction bounds (both upper and lower) for eigenfunctions. Eigenfunction restriction bounds have become an  an extremely active area in quantum chaos over the past several years, not only because of their intrinsic interest, but also because of their wide-ranging applications to the asymptotics of eigenfunction nodal domains and critical sets.**** In recent work with Zelditch and with El-Hajj,  I showed that indeed asymptotic eigenfunction restriction lower bounds along a real-analytic curve (ie. "goodness estimates") directly controls the intersection number of the eigenfunction nodal set with the curve. By making a judicious choice of curve,  in the case of arithmetic surfaces, Ghosh, Reznikov and Sarnak have recently shown one can inturn associated nodal domains with pairs of such intersection points. This leads to a very exciting new development in the field: a partial converse to the Courant nodal theorem. However, there are few cases where such quantitative lower bounds have been established.  Consequently, I propose to study the central questions:***Question 1: Given a curve $H$ on a Riemann surface $(M,g)$, under what conditions is $H$ necessarily a good curve?***Question 2: Under what conditions can one establish a quantitative lower bound in the Courant  theorem?***There are special cases that are of exceptional interest. Recently, with El-Hajj we answered Question 1 in the affirmative in the case where $H$ is strictly convex and the ambient manifold is a piecewise-smooth planar domain with ergodic bliiard dynamics. However, other cases remain completely open.  I propose to investigate both Questions 1 and 2 in other cases where the eigenfunction sequence is quantum ergodic. Finally, due to the high spectral multiplicity and the associated freedom in choosing eigenfunction bases, the case of general spherical harmonics should prove to be particularly fascinating and rich testing ground. With several of my students, I propose to investigate both questions in the latter case as well.*** *** *** **