Quantum Integrability and Inverse Spectral Theory

本文讨论了紧致黎曼流形谱的精细结构与其测地流动力学之间的关系。 一方面，首席研究员对众所周知的逆问题感兴趣，人们能听到鼓的形状吗？ 最近的进展被称为波迹不变量的光谱不变量的分析允许一个“听到"比以前可能的，特别是在某些真实的分析度量的情况下，完全可积测地线流。 这里的主要工具是建设和分析量子类似物的Birkhoff规范形式的哈密顿流附近的周期轨道或不变环面。另一方面，首席研究员也对与量子遍历性和混沌有关的数学物理问题感兴趣，或者在相反的极端，与完全可积性有关。 特别是，他希望分析对相关函数的频谱的一些模型量子映射在紧凑的辛流形和相关的精细结构不变量。 这些问题的动机起源于物理学。在五十年代，许多物理学家都在研究使原子核结合在一起的力。 这些力量是未知的，但它们的某些方面是可以观察到的。 特别是，原子核的能级谱可以在实验中测量。“逆谱问题”自然出现了--原子核的势能能否从这些能级中确定出来？ 答案基本上仍然是未知的。一方面，具有相同光谱的不同系统已经被构建;另一方面，这些模糊系统具有非常不寻常的特征，并且似乎典型系统很可能由它们的光谱决定。 另一个由核物理学家在50年代提出的问题，最著名的是E。Wigner和L.D.朗道的问题是高能级之间的间隔是否有任何模式。它们是随机发生的吗？ 这就是所谓的水平间距问题。许多计算机对物理系统和玩具数学模型的研究表明，模式与系统的可预测性或混沌程度有关。 为什么会这样，这仍然是一个神秘的数学问题。 然而，希望上述新工具和新发展将导致在深入了解这些基本问题方面取得有价值的收获。

This proposal is concerned with the relations between the fine structure of the spectrum of a compact Riemannian manifold and the dynamics of its geodesic flow. On the one hand, the principal investigator is interested in the well known inverse problem, can one hear the shape of a drum? Recent advances in the analysis of the spectral invariants known as wave trace invariants allow one to 'hear' much more than has been previously possible, especially in the case of certain real analytic metrics with completely integrable geodesic flow. The main tool here is the construction and analysis of quantum analogues of the Birkhoff normal forms for a Hamiltonian flow near a periodic orbit or invariant torus. On the other hand, the principal investigator is also interested in problems of mathematical physics having to do with quantum ergodicity and chaos or, at the opposite extreme, with complete integrability. In particular, he wishes to analyse pair correlation functions of the spectra of some model quantum maps over compact symplectic manifolds and related fine structure invariants. The motivation for these problems originated in physics. During the fifties, many physicists were studying the forces which hold the nucleus together. These forces were not known, but certain aspects of them were observable. In particular, the spectrum of energy levels of the nucleus could be measured in experiments. The 'inverse spectral problem' naturally arose- could the potential energy of the nucleus could be determined from these energy levels? The answer is still basically unknown. On the one hand, different systems with the same spectrum have been constructed; on the other, these ambiguous systems have very unusual features and it seems likely that typical systems are determined by their spectra. Another problem posed by nuclear physicists in the fifties, most famously by E. Wigner and L.D. Landau, was whether there was any pattern to the spacings between high energy levels. Do they just occur randomly? This is now known as the level spacings problem. Numerous computer studies of physical systems and toy mathematical models indicate that the patterns are related to the degree of predictability or chaos of the system. The reason why this should be so remains a mysterious mathematical problem. However, the hope is that the new tools and developments described above will lead to worthwhile gains in insight into these fundamental problems.