Inverse Anisotropic Problems and Resonances

Purdue University, West Lafayette IN

Investigators

Abstract

Proposal DMS-0400869 Title: Inverse anisotropic problems and resonances PI: Plamen Stefanov, Purdue University ABSTRACT The PI will work in the areas of Inverse Problems and Resonances. The PI will study mainly inverse problems of recovering a Riemannian metric from boundary or inverse scattering data. The central problem is the boundary rigidity problem for Riemannian metrics (called also inverse kinematic problem), where one has to recover a Riemannian metric in a bounded domain from the lengths of geodesics connecting every two boundary points. This problem is closely connected to the inverse problem of recovering a metric from the associated hyperbolic Dirichlet-to-Neumann (DN) map on the boundary, and also to an associated inverse spectral problem. The PI plans to study the problem of generic uniqueness for simple metrics and stability estimates. Related problems in integral geometry will be studied, like the linearized problem: recovery of tensors from integrals along geodesics and stability estimates. The PI will also work on the elliptic anisotropic inverse boundary value problem, the inverse backscattering problem for the acoustic equation, and related inverse problems where one has to recover the coefficients of the principal symbol of the differential operator from boundary or scattering data. In the area of Resonance Theory, the proposer plans to study the location and asymptotic distribution of resonances for various systems. Both scattering systems and semi-classical Schroedinger type of equations will be considered. Among the problems that will be studied are sharp upper bounds of the number of resonances in a disk or sector in the complex plane, upper and lower bounds connected with various characteristics of the trapped sets of the associated classical mechanics problem. Mathematical justification of numerical methods for computing resonances will be also considered. The properties of the scattering amplitude near resonances will be studied as well, which is related to the problem of observability of resonances. Inverse Problems, and in particular the problems in this proposal, is a mathematical tool of great importance to other sciences. Applications are numerous: they are used in medicine for imaging the internal structure of a human body and for medical diagnostics, in non-destructive material testing, in geophysics for obtaining information about the inner structure of the earth from seismic waves, in oil exploration, etc. Riemannian metric models anisotropic media, where the speed of wave propagation may depend not only on the position but also on the direction. The boundary rigidity problem is of interest not only to scattering theory but also to Riemannian geometry, its linearized version is a problem of independent interest in integral geometry as a generalized Radon transform. Resonance Theory is part of Scattering Theory for quantum mechanical and wave equation type of systems in unbounded domains. Resonances are certain frequencies that can be observed in a variety of situations in Quantum Chemistry, Physics, Acoustics, etc. Besides being motivated by applications in other natural sciences, Resonance Theory uses tools from and encourages further development of mathematics areas as Microlocal Analysis, Dynamical Systems, and Geometry.