Eigenfunction asymptotics and quantum chaos

本征函数渐进和量子混沌

基本信息

  • 批准号:
    RGPIN-2015-04979
  • 负责人:
  • 金额:
    $ 2.26万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2016
  • 资助国家:
    加拿大
  • 起止时间:
    2016-01-01 至 2017-12-31
  • 项目状态:
    已结题

项目摘要

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.
我的基本研究兴趣包括模拟电子密度概率分布的量子波函数的高能行为。我最感兴趣的是在高能极限下估计波函数振幅和频率的问题。   更精确地说,设$(M,g)$是紧致的,具有Laplace-Beltrami算子$\Delta_g的黎曼流形。$我提出的研究计划主要集中在相关的拉普拉斯特征函数的渐近性和相应特征值的Weyl渐近性上,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(上界和下界)上。本征函数约束界是近年来量子混沌研究中一个非常活跃的领域,不仅因为其内在的重要性,而且因为其在本征函数节域和临界集的渐近性方面的广泛应用。 在最近的工作与Zelditch和El-Hajj,我表明,确实渐近特征函数限制下限沿着一个真正的分析曲线(即。“优度估计”)直接控制特征函数节点集与曲线的交叉数。通过作出明智的选择曲线,在算术表面的情况下,Ghosh,Reznikov和Sarnak最近已经表明,人们可以在相关的节点域与这样的交点对。这导致了该领域一个非常令人兴奋的新发展:柯朗节点定理的部分匡威。然而,很少有这样的数量下限已经建立的情况下,因此,我建议研究的中心问题: 问题1:给定黎曼曲面$(M,g)$上的曲线$H$,在什么条件下$H$必然是好曲线? 问题2:在什么条件下可以建立柯朗定理的定量下限? 有一些特殊的情况是特别感兴趣的。最近,与El-Hajj,我们回答了肯定的情况下,$H$是严格凸的和周围的流形是一个分段光滑的平面域遍历blliard动力学的问题1。然而,其他情况下仍然完全开放。我建议调查的问题1和2在其他情况下,本征函数序列是量子遍历。最后,由于高光谱的多重性和相关的自由选择本征函数基地,一般球谐函数的情况下,应被证明是特别迷人的和丰富的试验场。与我的几个学生一起,我建议在后一种情况下也研究这两个问题。

项目成果

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Toth, John其他文献

Toth, John的其他文献

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{{ truncateString('Toth, John', 18)}}的其他基金

Eigenfunction Asymptotics and Quantum Chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2020-04700
  • 财政年份:
    2022
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction Asymptotics and Quantum Chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2020-04700
  • 财政年份:
    2021
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction Asymptotics and Quantum Chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2020-04700
  • 财政年份:
    2020
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics and quantum chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2015-04979
  • 财政年份:
    2019
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics and quantum chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2015-04979
  • 财政年份:
    2018
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics and quantum chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2015-04979
  • 财政年份:
    2017
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics and quantum chaos
本征函数渐进和量子混沌
  • 批准号:
    RGPIN-2015-04979
  • 财政年份:
    2015
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics on Riemannian manifolds
黎曼流形上的本征函数渐近
  • 批准号:
    170280-2010
  • 财政年份:
    2014
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics on Riemannian manifolds
黎曼流形上的本征函数渐近
  • 批准号:
    170280-2010
  • 财政年份:
    2013
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Eigenfunction asymptotics on Riemannian manifolds
黎曼流形上的本征函数渐近
  • 批准号:
    170280-2010
  • 财政年份:
    2012
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual

相似海外基金

Eigenfunction Asymptotics and Quantum Chaos
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  • 批准号:
    RGPIN-2020-04700
  • 财政年份:
    2022
  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
Asymptotics and singularity formation in Nonlinear PDEs related to fluid dynamic, geophysical flows, quantum physics and optics.
与流体动力学、地球物理流、量子物理和光学相关的非线性偏微分方程中的渐近和奇点形成。
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与流体动力学、地球物理流、量子物理和光学相关的非线性偏微分方程中的渐近和奇点形成。
  • 批准号:
    RGPIN-2019-06422
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本征函数渐进和量子混沌
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    RGPIN-2020-04700
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  • 资助金额:
    $ 2.26万
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本征函数渐进和量子混沌
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量子不变量的渐近
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    $ 2.26万
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与流体动力学、地球物理流、量子物理和光学相关的非线性偏微分方程中的渐近和奇点形成。
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  • 项目类别:
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本征函数渐进和量子混沌
  • 批准号:
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  • 资助金额:
    $ 2.26万
  • 项目类别:
    Discovery Grants Program - Individual
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与流体动力学、地球物理流、量子物理和光学相关的非线性偏微分方程中的渐近和奇点形成。
  • 批准号:
    RGPIN-2019-06422
  • 财政年份:
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  • 资助金额:
    $ 2.26万
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本征函数渐进和量子混沌
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