Irregularity and Gevrey Index Theorems for Linear Differential Operators

Abstract

In this chapter, the results of the preceding sections are applied to prove index theorems for linear differential operators in the spaces ${\Bbb C}[[x]]_s$ of s-Gevrey series as well as in the space ${\Bbb C}[[x]]_\infty = {\Bbb C}[[x]]$ of formal series and the space ${\Bbb C}[[x]]_0 = {\Bbb C}\{ x\}$ of convergent series, following a method by Deligne and Malgrange. The existence and value of the irregularity follow. An application to the Maillet-Ramis theorem which makes explicit the Gevrey order of solutions of linear ordinary differential equations is included. We also sketch a method based on wild analytic continuation, that is, continuation in the infinitesimal neighborhood.