Semi-Classical Analysis

半经典分析

• 批准号: 0200732
• 负责人: Maciej Zworski
• 金额: --
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2007-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200732&HistoricalAwards=false
• 关键词: Semi; Classical; Analysis

PI: Maciej R. Zworski, UC-Berkeley. DMS-0200732Abstract:The main interest of the PI is the study of quantum mechanics from themathematical point of view, and of its many manifestations in the theoryof partial differential equations and geometry. Specific current interests are the classical/quantum correspondence, resonances, geometric scattering, andnon-hermitian quantum mechanics. More precisely, the PI is interested in resonances, which are mathematical objects modeling states which have certain frequencies of oscillations (or rest energies) and rates of decay, such as unstable molecules or classical system responding to resonant forcing terms. Despite a long tradition and a lot of recent progress our understanding is still very limited. Current experimentaland numerical advances provide new stimuli for our studies. Another interest of the PI isnon-hermitian quantum mechanics, which deals with systems in which energy is not conserved.That is almost always present when we localize a part of a system and the global conservation of energy disapears. In a subtle way resonances already fall into this category of phenomena. The mathematical problems present here are the stability of eigenvalues and measuring the size of the resolvent of non-self-adjoint operators. That leads to the study of ``pseudospectra'' which is then related to many interesting phenomena in PDEs. Finally, the PI's interest involves mathematical scattering. It replaces spectral theory for problems on non-compact domains, and in physics almost all the data comes from scattering experiments. Many new things are constantly discovered now, ranging from scattering on locally symmetric spaces to problemsrelated to conformal field theory.Many of the phenomena discussed in this proposal are in fact more general: for instance, electromagnetic scattering can be used to model quantum scattering, and its understandingcan benefit from the development of the classical/quantum correspondence.The PI's work focuses on the search for general mathematical principles,and the detailed study of specific examples is motivated by that.

PI：Maciej R. Zworski，UC-Berkeley。 DMS-0200732ABSTRATCT：PI的主要兴趣是从主题的角度研究量子力学的研究，及其在部分微分方程和几何学理论中的许多表现。当前的特定兴趣是经典/量子对应关系，谐振，几何散射，non-Hermitian量子力学。更确切地说，PI对共振感兴趣，它们是具有某些具有振荡频率（或静息能）和衰减速率的数学对象，例如不稳定的分子或经典系统对谐振强迫术语做出响应。尽管有很长的传统和最近的许多进步，但我们的理解仍然非常有限。当前的实验和数值进步为我们的研究提供了新的刺激。 Pi的另一个兴趣是nonon-Hermitian量子力学，该机械师涉及无法保守能量的系统。当我们定位系统的一部分和全球能量消耗的系统时，几乎总是存在。以微妙的方式共鸣已经属于这种现象。此处存在的数学问题是特征值的稳定性和测量非自发接合运算符的分解的大小。这导致了对``pseudospectra''的研究，然后与PDE中的许多有趣现象有关。最后，PI的兴趣涉及数学散射。它取代了非紧缩域上问题的光谱理论，在物理学中，几乎所有数据都来自散射实验。现在不断发现许多新事物，从局部对称空间的散射到相关的田地理论，到该提议中讨论的现象的许多现象实际上更为笼统：例如，电磁散射可用于模拟量子散射，及其对普遍/量子的搜索的理解及其对普遍的搜索的理解。例子是由此激励的。