Inverse Anisotropic Problems and Resonances

提案DMS-0400869标题：反各向异性问题和共振PI：普渡大学普拉曼·斯特凡诺夫简介PI将在反问题和共振领域工作。PI将主要研究从边界或反向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的长度恢复有界域中的黎曼度量。这个问题与从边界上的双曲Dirichlet-to-Neumann映射恢复度量的反问题密切相关，也与相关的逆谱问题密切相关。PI计划研究简单度量和稳定性估计的一般唯一性问题。将研究积分几何中的相关问题，如线性问题：沿测地线积分的张量恢复和稳定性估计。PI还将处理椭圆各向异性反边值问题、声学方程的反后向散射问题以及相关的反问题，在这些反问题中，必须从边界或散射数据恢复微分算子的主符号的系数。在共振理论方面，作者计划研究各种系统的共振位置和渐近分布。散射系统和半经典薛定谔方程都将被考虑。在将要研究的问题中，包括复平面中圆盘或扇区中共振次数的精确上界，以及与相关经典力学问题的陷阱集的各种特征有关的上界和下界。还将考虑计算共振的数值方法的数学证明。还将研究共振附近的散射幅度的特性，这与共振的可观测性问题有关。逆问题，特别是这一建议中的问题，是对其他科学非常重要的数学工具。它们的应用非常广泛：用于医学成像人体内部结构和医学诊断，用于无损材料测试，用于地球物理从地震波获取有关地球内部结构的信息，在石油勘探等。黎曼度规模型用于各向同性介质，其中波的传播速度可能不仅取决于位置，而且还取决于方向。边界刚性问题不仅与散射理论有关，而且与黎曼几何有关，它的线性化形式是积分几何中一个独立于广义Radon变换的问题。共振理论是无界区域中量子力学和波动方程类系统的散射理论的一部分。共振是在量子化学、物理学、声学等各种情况下可以观察到的特定频率。除了在其他自然科学中的应用激励外，共振理论还使用并鼓励微局部分析、动力系统和几何等数学领域的进一步发展。

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.