Scattering and spectral theory for manifolds

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AbstractAward: DMS-0504729Principal Investigator: Harold DonnellyThe 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 nearthe ideal boundary of hyperbolic space.There is a largediscrepancy between the known upper and lower bounds for theresonance counting function of Euclidean space with odddimension.It has been conjectured that there is a general lowerbound which is comparable to the known upper bound.The proposalis to construct counterexamples which falsify theconjecture.Quantum unique ergodicity concerns concentration ofeigenfunctions on manifolds with ergodic geodesic flow.In earlierwork,manifolds were constructed where quantum unique ergodicityis violated for packets of eigenfunctions.It is now proposed toconstruct counterexamples to quantum unique ergodicity forindividual eigenfunctions.The proposed examples would beperturbations of manifolds with rotational symmetry,in theexceptional cases where KAM theory fails to provide invarianttori for the perturbed system.The Laplacian of hyperbolic spacehas absolutely continuous spectrum.If one alters the metric on acompact subset there may exist eigenfunctions whose eigenvalueslie below the bottom of the essential spectrum.The goal is tounderstand the nodal set of these eigenfunctions near the idealboundary at infinity.Resonances are mathematical models for physical phenomena relatedto atomic spectra.In general,states do not exist forever ifunperturbed and resonances correspond to decaying states whichoscillate.Mathematicians seek to understand quantum mechanics byanalogy with classical mechanics.If the classical motion ischaotic,one expects the probability distribution of the quantumparticle to be dispersed.Exceptions to this pattern,where theparticle concentrates,are of particular interest.The nodal set ofa quantum particle is the stationary set for the associated wavemotion.Its distribution and shape are of basic importance.