Inverse Anisotropic Problems and Resonances

dms -0400869题目：逆各向异性问题和共振espi: Plamen Stefanov，普渡大学摘要该PI将在逆问题和共振领域工作。pii将主要研究从边界或逆散射数据中恢复黎曼度量的反问题。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），其中必须从连接每两个边界点的测地线长度中恢复有界域中的黎曼度量。该问题与从边界上相关的双曲Dirichlet-to-Neumann （DN）映射中恢复度量的反问题以及相关的谱反问题密切相关。PI计划研究简单度量和稳定性估计的一般唯一性问题。相关的问题在积分几何将被研究，如线性化的问题：从沿测地线的积分恢复张量和稳定性估计。PI还将研究椭圆各向异性反边值问题，声学方程的反向散射问题，以及相关的反问题，其中必须从边界或散射数据中恢复微分算子主符号的系数。在共振理论方面，计划研究各种系统的共振的位置和渐近分布。散射系统和半经典薛定谔型方程都将被考虑。将研究的问题包括复平面中圆盘或扇形中共振数的尖锐上界，与相关经典力学问题的捕获集的各种特征相联系的上界和下界。计算共振的数值方法的数学论证也将被考虑。本文还研究了共振附近的散射振幅特性，这与共振的可观测性问题有关。反问题，特别是本文提出的问题，是对其他科学具有重要意义的数学工具。它们的应用非常广泛：在医学上用于人体内部结构的成像和医学诊断，在无损材料检测中，在地球物理学上用于从地震波中获取地球内部结构的信息，在石油勘探中，等等。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，而且取决于方向。边界刚性问题不仅是散射理论和黎曼几何感兴趣的问题，它的线性化版本作为广义Radon变换是积分几何中独立感兴趣的问题。共振理论是量子力学和无界域波动方程型系统散射理论的一部分。共振是在量子化学、物理、声学等多种情况下可以观察到的特定频率。除了受到其他自然科学应用的启发外，共振理论还使用了微局部分析、动力系统和几何等数学领域的工具，并鼓励这些领域的进一步发展。

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.