Huygens' Operators and Hadamard's Conjecture

线性双曲型微分算子的一个空区主要存在于传播锥内， 溶液完全消失。关于一般双曲型方程的缺项的研究是由I. G. Petrovsky（1945）开始的，Atiyah，Bott和Garding（1970-73）扩展和澄清了他的结果，发展了关于常系数双曲型算子的一个深刻而完整的理论。相比之下，对变系数算子的空隙知之甚少。本课题的主要目的之一是以推广Petrovsky-Atiyah-Bott-Garding理论的观点来研究这一经典问题。特别地，我们的努力旨在解决Hadamard关于Minkowski空间上满足Huygens原理的二阶解析波算子的显式确定的老问题。一些新的认识和问题（主要来自分析 和代数几何）所产生的与理论的空白也将进行探讨。波在连续介质中的传播问题是数学物理的基本问题之一，在自然科学和工程中有着重要的应用。特别感兴趣的（从实践和理论的角度来看）是波何时可以无扩散地传播以允许传输“清晰”（尖锐）信号的可能性的问题。在齐次空间中，这个问题已经得到了很好的研究，但在很大程度上仍然是开放的。我的项目旨在开发新的数学工具和技术来研究非均匀和各向异性介质中的这一难题。所寻求的结果在波传播的数学理论中具有基本的兴趣和意义，并可能在相关的物理学科中有应用，包括电磁波和声波理论、空间通信技术、磁流体力学、晶体光学等。

ABSTRACT A lacuna of a linear hyperbolic differential operator is adomain inside the propagation cone where its fundamental solution vanishes identically. The study of lacunas for generalhyperbolic equations was initiated by I.G.Petrovsky (1945).Extending and clarifying his results, Atiyah, Bott and Garding(1970-73) developed a profound and complete theory for hyperbolicoperators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. The investigation of this classical problem with a view ofgeneralizing the Petrovsky-Atiyah-Bott-Garding theory is one of the primary purposes of my project. In particular, theefforts will be aimed at resolving the old question ofJ.Hadamard on explicit determination of second order analyticwave operators satisfying Huygens' principle on Minkowskispaces. Some new conjectures and problems (mostly from Analysis and Algebraic Geometry) arising in connection with the theoryof lacunas will be also explored. The study of propagation of waves in continuous media is one ofthe fundamental problems in Mathematical Physics with many important applications in natural sciences and engineering. Of special interest (both from practical and theoretical point of view) is the question of when the waves may propagate without diffusion to allow the possibility of transmitting `clean-cut' (sharp) signals. Well-studied in homogeneous spaces this question remains largely open in general. My project aims to develop new mathematical tools and techniques to investigate this difficult problem in the case ofinhomogeneous and anisotropic media. The results sought are of fundamentalinterest and significance in mathematical theory of wave propagation and may have applications in related physical disciplines including thetheory of electromagnetic and acoustic waves, space communicationtechnologies, magnetohydrodynamics, crystal optics, etc.