椭圆算子的可逆二重
The invertible double of elliptic operators
摘要：
首先，我们回顾了狄拉克算子的民间传说的狄拉克型算子的基本分析和几何性质的紧致流形光滑边界和封闭的分区流形，并显示这些属性如何依赖于建设一个典型的可逆双和有关的概念的Calderon投影。然后，我们总结了一个最近的构造一般的一阶椭圆型微分算子在光滑紧流形上的边界的一个典型的可逆双。我们推导出一个自然公式的Calderon投影产生一个推广的著名的Cobordism定理。我们提供了一个列表的假设，以获得一个连续变化的卡尔德龙投影下的平滑变化的系数。这就产生了各种新的谱流定理。最后，我们勾画了一个研究方案，分别封闭几何狄拉克算子类型情况和一般线性椭圆情况之间的最后剩余差距。
英文摘要：
First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderon projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case.
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