Quantum Dynamics: Geometry and Spectrum

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AbstractAward: DMS-0071358Principal Investigator: Steve ZelditchWe propose to apply quantum dynamical notions to problems ingeometry, inverse spectral theory and mathematical physics.First is the well-known problem, `Can you hear the shape of ananalytic drum?' We have constructed a `quantum Birkhoff normalform' of the Laplacian around a bouncing ball orbit and haveproved that it is a spectral invariant. We will investigate theextent to which a bounded simply connected analytic plane domainis determined by its normal forms at one or more bouncing ballorbits. We will also investigate a new connection between thespectrum and X-ray tomography of the domain. The second group ofproblems involves applications of a recently developedprobabalistic theory of holomorphic sections of line bundles overKahler manifolds. We have at this time determined much of thelocal statistics of sections over small balls. We have alsogeneralized the results to symplectic manifolds. We now aim toapply these results to determine hole probabilities,probabilities of transversality, and other matters which arise incomplex and symplectic geometry. The third group of problemsbelongs to Quantum Chaos. We have introduced a new model ofrandom quantum maps, obtained by quantizing stochasticHamiltonian flows. We propose to investigate the main problemsof Quantum Chaos for this model.Much of physics of the twentieth century has revolved around thequantum mechanics of particles and fields, in particular aroundthe Schrodinger equation, and the physics and engineering of thefuture will be even more quantum mechanical. Quantum systems arequite un-intuitive and one of the best ways to understand whatthey do is to compare them to the classical systems which theyquantize. For instance, a hydrogen atom in a strong magneticfield can be thought of as a small particle moving in a forcefield. As the magnetic field gets stronger, the particle'smotion becomes more complex and chaotic. What does this say aboutthe quantum particle? According to twenty years of numericalstudies, the behaviour of a chaotic quantum particle should berandom, with statistics similar to those for random matrices.Why should solutions of the Schrodinger equation know anythingabout random matrices? Our approach is to introduce a new modelof chaotic quantum systems which allows one to consider largefamilies of systems at once. In a sense, we are trying toshow that a `random' quantum system behaves similarly to a`random' matrix, despite the differences in these systems.Such a model should be able to explain observations ranging fromnuclear resonance levels to chemical reaction rates.