Inverse Anisotropic Problems and Resonances

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Proposal DMS-0400869Title: Inverse anisotropic problems and resonancesPI: Plamen Stefanov, Purdue UniversityABSTRACTThe PI will work in the areas of Inverse Problems and Resonances. The PIwill study mainly inverse problems of recovering a Riemannian metric fromboundary or inverse scattering data. The central problem is the boundaryrigidity problem for Riemannian metrics (called also inverse kinematicproblem), where one has to recover a Riemannian metric in a bounded domainfrom the lengths of geodesics connecting every two boundary points. Thisproblem is closely connected to the inverse problem of recovering a metricfrom the associated hyperbolic Dirichlet-to-Neumann (DN) map on theboundary, and also to an associated inverse spectral problem. The PI plansto study the problem of generic uniqueness for simple metrics and stabilityestimates. Related problems in integral geometry will be studied, like thelinearized problem: recovery of tensors from integrals along geodesics andstability estimates. The PI will also work on the elliptic anisotropicinverse boundary value problem, the inverse backscattering problem for theacoustic equation, and related inverse problems where one has to recover thecoefficients of the principal symbol of the differential operator fromboundary or scattering data. In the area of Resonance Theory, the proposerplans to study the location and asymptotic distribution of resonances forvarious systems. Both scattering systems and semi-classical Schroedingertype of equations will be considered. Among the problems that will bestudied are sharp upper bounds of the number of resonances in a disk orsector in the complex plane, upper and lower bounds connected with variouscharacteristics of the trapped sets of the associated classical mechanicsproblem. Mathematical justification of numerical methods for computingresonances will be also considered. The properties of the scatteringamplitude near resonances will be studied as well, which is related to theproblem of observability of resonances. Inverse Problems, and in particular the problems in this proposal, is amathematical tool of great importance to other sciences. Applications arenumerous: they are used in medicine for imaging the internal structure of ahuman body and for medical diagnostics, in non-destructive material testing,in geophysics for obtaining information about the inner structure of theearth from seismic waves, in oil exploration, etc. Riemannian metric modelsanisotropic media, where the speed of wave propagation may depend not onlyon the position but also on the direction. The boundary rigidity problem isof interest not only to scattering theory but also to Riemannian geometry,its linearized version is a problem of independent interest in integralgeometry as a generalized Radon transform. Resonance Theory is part ofScattering Theory for quantum mechanical and wave equation type of systemsin unbounded domains. Resonances are certain frequencies that can beobserved in a variety of situations in Quantum Chemistry, Physics,Acoustics, etc. Besides being motivated by applications in other naturalsciences, Resonance Theory uses tools from and encourages furtherdevelopment of mathematics areas as Microlocal Analysis, Dynamical Systems,and Geometry.