Inverse Anisotropic Problems and Resonances

提案DMS-0400869题目：逆各向异性问题和共振PI：Plamen Stefanov，普渡大学摘要PI将在逆问题和共振领域工作。PI将主要研究从边界或逆散射数据恢复黎曼度量的逆问题。中心问题是黎曼度量的边界刚性问题（也称为逆运动学问题），即在有界区域中由连接边界点的测地线的长度恢复黎曼度量。这个问题与从边界上的双曲Dirichlet-to-Neumann（DN）映射中恢复度量的逆问题密切相关，也与一个逆谱问题密切相关。PI计划研究简单度量和稳定性估计的通用唯一性问题。将研究积分几何中的相关问题，如线性化问题：从沿着测地线的积分恢复张量和稳定性估计。PI还将工作的椭圆各向异性逆边值问题，反向散射问题的声学方程，以及相关的逆问题，其中一个必须恢复thecoefficients的主要符号的微分算子从边界或散射数据。在共振理论方面，作者计划研究各种系统共振的位置和渐近分布。散射系统和半经典Schroedinger型方程都将被考虑。其中的问题，将被研究的是尖锐的上限的数量的共振盘或扇区在复杂的平面，上限和下限连接的各种特征的陷阱集的相关经典mechanicsproblem。计算共振的数值方法的数学理由也将被考虑。研究了共振附近散射振幅的性质，这与共振的可观测性问题有关。反问题，特别是本文提出的反问题，是对其他学科有重要意义的数学工具。应用程序繁多：它们在医学中用于对人体的内部结构进行成像，并用于医学诊断，用于非破坏性材料测试，用于从地震波中获得关于地球内部结构的信息的物理学，用于石油勘探等。黎曼度量模型是各向同性介质，其中波的传播速度不仅取决于位置，还取决于方向。边界刚性问题不仅是散射理论所关心的问题，而且也是黎曼几何所关心的问题，它的线性化形式作为广义Radon变换是积分几何中一个独立的问题。共振理论是无界域中量子力学和波动方程类型系统的散射理论的一部分。共振是某些频率，可以在量子化学，物理学，声学等各种情况下观察到，除了在其他自然科学中的应用，共振理论使用的工具，并鼓励进一步发展的数学领域，如微观分析，动力系统和几何。

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.