Stokes matrices of a reducible equation with two irregular singularities of Poincaré rank 1 via monodromy matrices of a reducible Heun type equation

Abstract

We consider a second order reducible equation having non-resonant irregular singularities at $x=0$ and $x=\infty$. Both of them are of Poincaré rank 1. We introduce a small complex parameter $\epsilon$ that splits together $x=0$ and $x=\infty$ into four different Fuchsian singularities $x_L=-\sqrt{\epsilon },x_R=\sqrt{\epsilon }$ and $x_{LL}=-1/\sqrt{\epsilon },x_{RR}=1/\sqrt{\epsilon }$, respectively. The perturbed equation is a second order reducible Fuchsian equation with 4 different singularities, i.e. a Heun type equation. Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial equation are realized as a limit of the nilpotent parts of the monodromy matrices of the perturbed equation when $\epsilon \to 0$ in the real positive direction. To establish this result we combine a direct computation with a theoretical approach.

References in corpus (1)

• Stokes matrices for a class of reducible equations