Quantum Integrability and Inverse Spectral Theory

本文研究了紧致黎曼流形谱的精细结构与其测地线流动力学之间的关系。一方面，首席研究员对著名的反问题很感兴趣，有人能听到鼓的形状吗？谱不变量（即波迹不变量）分析的最新进展使人们能够“听到”比以前可能的更多，特别是在具有完全可积测地线流的某些真实解析度量的情况下。这里的主要工具是构造和分析在周期轨道或不变环面附近的哈密顿流的Birkhoff范式的量子类似物。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。特别地，他希望分析紧辛流形上某些模型量子映射的谱的对相关函数及其精细结构不变量。这些问题的动机源于物理学。在五十年代，许多物理学家都在研究使原子核凝聚在一起的力。这些力是未知的，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。“逆光谱问题”自然产生了——能否从这些能级中确定原子核的势能？答案基本上仍然是未知的。一方面，构建了具有相同光谱的不同系统；另一方面，这些模糊系统具有非常不寻常的特征，似乎典型的系统是由它们的光谱决定的。50年代核物理学家提出的另一个问题，最著名的是E.维格纳（E. Wigner）和L.D.朗道（L.D. Landau），是高能能级之间的间隔是否存在某种模式。它们是随机发生的吗？这就是所谓的水平间隔问题。对物理系统和数学模型的大量计算机研究表明，这些模式与系统的可预测性或混乱程度有关。为什么会这样，这仍然是一个神秘的数学问题。然而，我们希望上面描述的新工具和发展将在洞察这些基本问题方面带来有价值的收获。

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.