Semi-Classical Analysis

PI：Maciej R.兹沃斯基加州大学伯克利分校DMS-0200732摘要：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和几何理论中的许多表现形式。目前的兴趣是经典/量子对应，共振，几何散射，和非厄米量子力学。更确切地说，PI对共振感兴趣，共振是对具有特定振荡频率（或静止能量）和衰变率的状态进行建模的数学对象，例如不稳定分子或响应共振强迫项的经典系统。尽管有着悠久的传统和最近取得的许多进展，但我们的理解仍然非常有限。目前的实验和数值计算的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而整体能量守恒消失时，几乎总是存在。共振以一种微妙的方式已经属于这一类现象。这里提出的数学问题是本征值的稳定性和测量的大小非自伴算子的预解式。这导致了对“伪谱”的研究，它与偏微分方程中许多有趣的现象有关。最后，PI的兴趣涉及数学散射。它取代了非紧域问题的谱理论，在物理学中几乎所有的数据都来自散射实验。从局部对称空间上的散射到与共形场论有关的问题，现在有许多新的东西不断被发现。在这个提议中讨论的许多现象实际上是更普遍的：例如，电磁散射可以用来模拟量子散射，它的理解可以受益于经典/量子对应的发展。PI的工作集中在寻找一般的数学原理，对具体例子的详细研究也是出于此目的。

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.