Eigenfunction asymptotics and quantum chaos

本征函数渐进和量子混沌

• 批准号: RGPIN-2015-04979
• 负责人: Toth, John
• 金额: $ 2.26万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=683846
• 关键词: Eigenfunction; asymptotics; quantum; chaos

***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.*** *** *** **

***我的基础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。  我最感兴趣的是估计高能极限下的波函数振幅和频率的问题。*** *** 更准确地说，让 $(M,g)$ 是带有 Laplace-Beltrami 算子的紧凑黎曼流形 $\Delta_g。$ 我提出的研究计划重点是相关拉普拉斯本征函数和 Weyl 渐近函数的渐进性 相应的特征值。  我的研究计划的很大一部分集中在特征函数的渐近限制界（上限和下限）上。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极其活跃的领域，不仅因为它们的内在兴趣，还因为它们在本征函数节点域和临界集的渐进性方面的广泛应用。****在最近与 Zelditch 和 El-Hajj 的合作中，我证明了沿着实解析的渐近本征函数限制下界确实存在 曲线（即“优度估计”）直接控制特征函数节点集与曲线的交集数。在算术曲面的情况下，通过明智地选择曲线，Ghosh、Reznikov 和 Sarnak 最近表明，可以将相关的节点域与成对的此类交点进行关联。这导致了该领域一个非常令人兴奋的新发展：与库朗节点定理的部分逆向。然而，建立这种定量下限的情况很少。  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 $H$ 一定是一条好曲线？***问题 2：在什么条件下可以建立库朗定理中的定量下界？***有些特殊情况特别令人感兴趣。最近，在 El-Hajj 中，我们在 $H$ 严格凸且环境流形是具有遍历 bliiard 动力学的分段光滑平面域的情况下对问题 1 做出了肯定的回答。然而，其他案件仍然完全悬而未决。  我建议在本征函数序列是量子遍历的其他情况下研究问题 1 和问题 2。最后，由于高谱多重性和选择本征函数基的相关自由度，一般球谐函数的情况应该被证明是特别令人着迷和丰富的试验场。我建议与我的几位学生一起研究后一种情况中的这两个问题。*** *** *** **