Semi-Classical Analysis

派：Maciej R.Zworski，加州大学伯克利分校。PI的主要兴趣是从数学的角度研究量子力学，以及它在偏微分方程组和几何理论中的许多表现。目前特别感兴趣的是经典/量子对应、共振、几何散射和非厄米量子力学。更准确地说，PI对共振感兴趣，共振是对具有一定振荡频率(或静止能量)和衰减率的状态进行建模的数学对象，例如不稳定的分子或响应共振强迫项的经典系统。尽管有悠久的传统和最近的许多进展，但我们的理解仍然非常有限。目前的实验和数值进展为我们的研究提供了新的刺激。PI的另一个兴趣是非厄米量子力学，它处理的是能量不守恒的系统。当我们局部化系统的一部分，全局能量守恒消失时，这几乎总是存在的。以一种微妙的方式，共振已经属于这一类现象。这里提出的数学问题是特征值的稳定性和非自伴算子预解的大小的度量。这导致了对“伪谱”的研究，这与偏微分方程组中的许多有趣现象有关。最后，PI的兴趣涉及数学上的分散。它取代了谱理论来解决非紧致域上的问题，在物理学中，几乎所有的数据都来自散射实验。现在不断发现许多新事物，从局部对称空间上的散射到与保形场理论相关的问题。实际上，这个提议中讨论的许多现象都更普遍：例如，电磁散射可以用来模拟量子散射，它的理解可以从经典/量子对应的发展中受益。PI的工作集中在寻找一般的数学原理上，对具体例子的详细研究受到了这一点的启发。

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.