Principal Investigator: Harold Donnelly
The principal investigator plans to study three separate topics: i) Lower bounds on the counting function for resonances, ii) Quantum unique ergodicity, iii) Behavior of eigenfunctions near the ideal boundary of hyperbolic space.There is a large discrepancy between the known upper and lower bounds for the resonance counting function of Euclidean space with odd dimension.It has been conjectured that there is a general lower bound which is comparable to the known upper bound.The proposal is to construct counterexamples which falsify the conjecture.Quantum unique ergodicity concerns concentration of eigenfunctions on manifolds with ergodic geodesic flow.In earlier work,manifolds were constructed where quantum unique ergodicity is violated for packets of eigenfunctions.It is now proposed to construct counterexamples to quantum unique ergodicity for individual eigenfunctions.The proposed examples would be perturbations of manifolds with rotational symmetry,in the exceptional cases where KAM theory fails to provide invariant tori for the perturbed system.The Laplacian of hyperbolic space has absolutely continuous spectrum.If one alters the metric on a compact subset there may exist eigenfunctions whose eigenvalues lie below the bottom of the essential spectrum.The goal is to understand the nodal set of these eigenfunctions near the ideal boundary at infinity.
Resonances are mathematical models for physical phenomena related to atomic spectra.In general,states do not exist forever if unperturbed and resonances correspond to decaying states which oscillate.Mathematicians seek to understand quantum mechanics by analogy with classical mechanics.If the classical motion is chaotic,one expects the probability distribution of the quantum particle to be dispersed.Exceptions to this pattern,where the particle concentrates,are of particular interest.The nodal set of a quantum particle is the stationary set for the associated wave motion.Its distribution and shape are of basic importance.
