一阶自伴Shapiro-Lopatinskii边值问题的指标
The index of self-adjoint Shapiro-Lopatinskii boundary problems of order one
AI总结：
本文研究1阶自伴Shapiro-Lopatinskii边值问题族的Atiyah-Bott-Singer指标定理类似形式，基于拓扑与伪微分方法，克服自伴情形下的实现障碍，并推广了Gorokhovsky和Lesch的结果。
摘要：
本文致力于研究1阶自伴椭圆（即满足Shapiro-Lopatinskii条件）局部边值问题族的Atiyah-Bott-Singer指标定理的类似形式。证明基于经典的拓扑和伪微分方法，但在自伴情形下会遇到一些新现象。拓扑指标遵循Atiyah-Bott的方法定义，但在自伴情形下会遇到经典情况中不存在的障碍。解析指标借助作者在arXiv:2111.15081中推广Atiyah-Singer方法的研究来定义。在解析指标方面，会遇到用自伴边值问题实现符号的障碍，这与定义拓扑指标的障碍类似。作为应用，我们推广了Gorokhovsky和Lesch（arXiv:1310.0210）的结果。本文第一版仅在附加技术假设下证明了该指标定理。第二版中发展的符号与算子乘法性质理论，使得该假设得以去除。
英文摘要：
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.
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