Degeneration of the Stokes Geometry for Higher-Order Equations
摘要
In Chap. 1 we observed that there are two kinds of crossing points of three Stokes curves, that is, trilemma-type crossing points and triordered-type crossing points. Although the configuration of Stokes curves are similar between these two kinds of crossing points, they have quite different characters. In this chapter we discuss their difference from the viewpoint of the degeneration of the Stokes geometry. As explained in Honda et al. (Virtual Turning Points. Springer Briefs in Mathematical Physics, vol. 4. Springer-Verlag, Berlin, 2015, Chapter 2), degeneration of the Stokes geometry for higher-order equations is described by “growing trees” and a trilemma-type crossing point is the simplest example of such growing trees. Furthermore, degeneration of the Stokes geometry is related to the location of singularities of Borel transformed WKB solutions via the so-called periods. Using the \((1,4)\) hypergeometric system discussed in Sect. 1.5.4 as an example, we will confirm this intriguing fact by investigating its bicharacteristic curve concretely. In contrast with this, a triordered-type crossing point has no relationship with degeneration of the Stokes geometry.