Quantum Integrability and Inverse Spectral Theory

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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.