On the exact WKB analysis of second order linear ordinary differential equations with simple poles (Q1581084)

Consider the linear differential equation \[ \Bigl(-\frac{d^2}{dx^2}+\eta^2\Bigl(\frac{Q_0(x)}{x}+ \eta^{-2}\frac{Q_2(x)}{x^2}\Bigr)\Bigr)\psi=0 \tag{E} \] near the origin. Here, \(Q_0(x)\) and \(Q_2(x)\) are holomorphic functions near the origin satisfying \(Q_0(0)\not=0,\) and \(\eta\) is a large parameter. Here, the author shows the Borel summability of WKB solutions to (E) which has the singularity of square-root type at the origin. Furthermore, the connection formula for the Borel transform of WKB solutions near a simple pole is studied.