论文标题
自我接合shapiro-lopatinskii的索引命令的边界问题
The index of self-adjoint Shapiro-Lopatinskii boundary problems of order one
论文作者
论文摘要
该论文致力于对Atiyah-Bott-Singer索引定理的类似物，用于自我拥护者椭圆家族（即满足Shapiro-lopatinskii条件）的本地边界问题1的本地边界问题。证明是基于经典拓扑结构和伪差异方法，但在自我偶发方法中，但在自我偶发中有一些新现象。拓扑指数是在Atiyah-Bott之后定义的，但是在自我伴侣案例中，在经典情况下遇到了不存在的阻塞。分析指数是在作者的方法Arxiv：2111.15081的帮助下定义的，该方法概括了Atiyah-Singer之一。在分析指数方面，一个人遇到了通过自我接合边界问题实现符号的障碍，类似于定义拓扑指数的障碍。作为应用程序，我们概括了Gorokhovsky和Lesch Arxiv的结果：1310.0210。 在本文的第一个版本中，索引定理仅在额外的技术假设下才得到证明。在第二版中开发的符号和运算符的乘法属性的理论允许删除此假设。
The paper is devoted to an analogue of Atiyah-Bott-Singer index theorem for families of self-adjoint elliptic (i.e. satisfying the Shapiro-Lopatinskii condition) local boundary problems of order 1. The proofs are based on classical topological and pseudo-differential methods, but in the self-adjoint case one encounters some new phenomena. The topological index is defined following Atiyah-Bott, but in the self-adjoint case one encounters an obstruction not present in the classical situation. The analytical index is defined with the help of author's approach arXiv:2111.15081, which generalized the one of Atiyah-Singer. On the analytic index side one encounters an obstruction to the realization of symbols by self-adjoint boundary problems, similar to the obstruction to defining the topological index. As an application, we generalize results of Gorokhovsky and Lesch arXiv:1310.0210. In the first version of this paper the index theorem was proved only under an additional technical assumption. A theory of multiplicative properties of symbols and operators, developed in the second version, allows to remove this assumption.
