Stokes matrices of a reducible equation with two irregular singularities of Poincar\'{e} rank 1 via monodromy matrices of a reducible Heun type equation

TL;DR

This paper investigates the Stokes matrices of a reducible second-order differential equation with two irregular singularities of Poincaré rank 1, showing they can be obtained as limits of monodromy matrices of a perturbed Heun-type equation with four singularities.

Contribution

It introduces a perturbation approach that links the Stokes matrices of the original equation to monodromy matrices of a related Fuchsian equation, providing a new method to analyze irregular singularities.

Findings

Stokes matrices are limits of monodromy matrices as perturbation parameter tends to zero.

The perturbed equation is a reducible Fuchsian (Heun type) equation with four singularities.

The approach combines direct computation with theoretical analysis.

Abstract

We consider a second order reducible equation having non-resonant irregular singularities at $x=0$ and $x=\infty$. Both of them are of Poincar\'{e} rank 1. We introduce a small complex parameter $\varepsilon$ that splits together $x=0$ and $x=\infty$ into four different Fuchsian singularities $x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}$ and $x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$, 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 $\varepsilon \rightarrow 0$ in the real positive direction. To establish this result we combine a direct computation with a theoretical approach.