FRG Collaborative Proposal: Eigenfunctions of the Laplacian

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Collaborative proposal DMS 0354386/0354539/0354668PIs: C.Sogge, S.Zelditch (Johns Hopkins)/D.Tataru, M.Zworski(U California, Berkeley)/H.Smith (U Washington)Title: Eigenfunctions of the LaplacianABSTRACTThis proposal is concerned with estimates of solutions of waveequations on (both compact and non-compact) Riemannian manifolds, possibly with boundary. We are interested in how thegeometry, the boundary and the regularity of the metric influencecertain basic estimates. Problems of this kind arise both in thestudy of local and global existence of solutions of nonlinear waveequations and in the study of eigenfunctions of Laplacians inquantum chaos. Although these topics are widely separated in theirphysical origins, the relevant mathematics is closely related andthe techniques and insights in the two areas cross fertilize ina fruitful way. Each of the PIs is an expert one of these areas.By pooling our knowledge through a FRG grant we shall be able to make significant progress in these fields as wellas possibly in related ones such as general relativity theory.Eigenfunctions are the building blocks of functions. Understandingtheir basic size and concentration properties helps us solve new differential equations. In the other direction, techniques fromthe study of nonlinear wave equations have recently been usedto prove new results about eigenfunctions. This proposal isabout continuing this cross-fertilization of related fields.