We describe the exact WKB method from the point of view of abelianization, both for Schrodinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties. We also briefly revisit the Schrodinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.

中文翻译：

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$$T_3$$ 方程的精确 WKB 和阿贝尔化

我们从阿贝尔化的角度描述了精确的 WKB 方法，包括薛定谔算子和它们的高阶类似物 (opers)。我们考虑的主要新示例是“$T_3$ 方程”，这是三次穿孔球体上的阶 $3$ 方程，在穿孔处具有规则奇点。在这种情况下，精确的 WKB 分析导致在平坦的 $\mathrm{SL}(3)$-连接的模空间上考虑一种新的 Darboux 坐标系。我们给出这样一个坐标系的最简单的例子，并用数值验证在这些坐标中，$T_3$方程的单向性具有预期的渐近性质。我们还从阿贝尔化的角度简要回顾了具有三次势能的薛定谔方程和马修方程。