Nonlocal Boundary Value Problems: Index Theory and Semiclassical Asymptotics

非局部问题出现在数学及其在科学和技术中的应用的许多领域。在这个项目中，我们感兴趣的是与流形上的群作用相关的非局部边值问题。在这些问题中，主算子和边界算子都属于由流形上的(伪)微分算子生成的算子代数，以及定义群作用的与微分同胚有关的所谓的移位算子。这种类型的非局域问题出现在等离子体物理、航空航天中的多层板壳理论、具有二维反馈的光学系统等中。从非对易几何的观点来看，它们也是有趣的，因为非局域算子的符号本质上形成了非对易代数。我们打算研究与这种群作用相关的椭圆和双曲非局部边值问题。我们的目的是(I)研究理论的分析方面(即引入椭圆性的概念并证明椭圆元的Fredholm性)，(Ii)应用拓扑学和非对易几何的方法来建立指数公式，以及(Iii)使用半经典方法来获得双曲问题的渐近性。

Nonlocal problems arise in many areas of mathematics and its applications to science and technology. In this project, we are interested in nonlocal boundary value problems associated with group actions on manifolds. In these problems, both the main operator and the boundary operator belong to operator algebras generated by (pseudo)differential operators on the manifold and so-called shift operators associated with diffeomorphisms, which define a group action. Nonlocal problems of this type arise in plasma physics, in the theory of multi-layer plates and shells used in aviation and astronautics, in optical systems with two-dimensional feedback, etc. They are also interesting from the point of view of noncommutative geometry, because symbols of nonlocal operators form essentially noncommutative algebras. We intend to study elliptic and hyperbolic nonlocal boundary value problems associated with such group actions. Our aim is (i) to investigate the analytical aspects of the theory (i.e. to introduce a notion of ellipticity and to prove the Fredholm property of elliptic elements), (ii) to apply methods of topology and noncommutative geometry in order to establish index formulas and (iii) to use semiclassical methods to obtain asymptotics in hyperbolic problems.