通过可约Heun型方程的单值矩阵可约的Poincaré秩为1的具有两个非正则奇点的方程的Stokes矩阵
Stokes matrices of a reducible equation with two irregular singularities of Poincaré rank 1 via monodromy matrices of a reducible Heun type equation
摘要：
考虑一类在$x=0$和$x=\infty$处具有非共振非正则奇点的二阶可约方程.它们都是庞加莱秩1。我们引入了一个小的复杂参数$\varepsilon$，它将$x=0$和$x=\infty$分别分成四个不同的Fuchsian奇点$x_L=-\sqrt{\varepsilon}, x_R=\sqrt{\varepsilon}$和$x_{LL}=-1/\sqrt{\varepsilon}, x_{RR}=1/\sqrt{\varepsilon}$。扰动方程是一个具有4个不同奇点的二阶可约Fuchsian方程，即Heun型方程.然后证明了当扰动方程恰好有两个不同类型的共振奇点时，当$\varepsilon \rightarrow 0$在实正方向时，初始方程的所有Stokes矩阵都可实现为扰动方程的monodromy矩阵的幂零部分的极限.为了建立这个结果，我们结合了直接计算与理论方法。
英文摘要：
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 $\varepsilon$ 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.
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