On the complex WKB analysis for a Schrödinger equation with a general three-segment characteristic polygon (Q1411132)

Consider a linear differential equation of the form \[ \begin{aligned} &\varepsilon^{2h} \frac{d^2y}{dx^2}= a(x,\varepsilon)y, \qquad 0\leq | x| \leq x_0, \quad 0<\varepsilon\leq \varepsilon_0, \\ &a(x,\varepsilon):= \sum^k_{j=0}a_j\varepsilon^j x^{h+m+k-2j}+ \sum^h_{j=k+1}a_j\varepsilon^jx^{h+m-j}, \quad a_j\not=0, \end{aligned} \] \(h\geq 3\); \(1\leq k\leq h-1\); \(0\leq m\leq h-3,\) which admits a turning point at \(x=0.\) In this case, the corresponding characteristic polygon has three segments. The purpose of this paper is to clarify the behaviour of solutions around the turning point. In appropriate subdomains, the Stokes curve configurations for the reduced equations are constructed, and by the use of WKB analysis, the matching matrices are calculated.