Quantum Dynamics: Geometry and Spectrum

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Abstract Award: DMS-0071358 Principal Investigator: Steve Zelditch We propose to apply quantum dynamical notions to problems in geometry, inverse spectral theory and mathematical physics. First is the well-known problem, `Can you hear the shape of an analytic drum?' We have constructed a `quantum Birkhoff normal form' of the Laplacian around a bouncing ball orbit and have proved that it is a spectral invariant. We will investigate the extent to which a bounded simply connected analytic plane domain is determined by its normal forms at one or more bouncing ball orbits. We will also investigate a new connection between the spectrum and X-ray tomography of the domain. The second group of problems involves applications of a recently developed probabalistic theory of holomorphic sections of line bundles over Kahler manifolds. We have at this time determined much of the local statistics of sections over small balls. We have also generalized the results to symplectic manifolds. We now aim to apply these results to determine hole probabilities, probabilities of transversality, and other matters which arise in complex and symplectic geometry. The third group of problems belongs to Quantum Chaos. We have introduced a new model of random quantum maps, obtained by quantizing stochastic Hamiltonian flows. We propose to investigate the main problems of Quantum Chaos for this model. Much of physics of the twentieth century has revolved around the quantum mechanics of particles and fields, in particular around the Schrodinger equation, and the physics and engineering of the future will be even more quantum mechanical. Quantum systems are quite un-intuitive and one of the best ways to understand what they do is to compare them to the classical systems which they quantize. For instance, a hydrogen atom in a strong magnetic field can be thought of as a small particle moving in a force field. As the magnetic field gets stronger, the particle's motion becomes more complex and chaotic. What does this say about the quantum particle? According to twenty years of numerical studies, the behaviour of a chaotic quantum particle should be random, with statistics similar to those for random matrices. Why should solutions of the Schrodinger equation know anything about random matrices? Our approach is to introduce a new model of chaotic quantum systems which allows one to consider large families of systems at once. In a sense, we are trying to show 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 from nuclear resonance levels to chemical reaction rates.