Shatalov-Sternin's construction of complex WKB solutions and the choice of integration paths (Q2929425)

The author discusses methods of resurgent analysis to study the asymptotics of solutions of linear ODE with a small parameter. An equation of the type NEWLINE\[NEWLINE\left( 1 \right)\quad -{{\varepsilon }^{2}}\partial _{x}^{2}\varphi (\varepsilon ,\text{ }x)\text{ }+\text{ }V(x)\varphi (\varepsilon ,\text{ }x)=0,\quad NEWLINE\]NEWLINE where \(\varepsilon \) is a small complex parameter, and \(V(x)\) is an entire function, can be transformed by the Laplace transformation \(\int_{\gamma }{\Phi (s){{e}^{-{s}/{\varepsilon }\;}}ds}\) into an equation NEWLINE\[NEWLINE\partial _{x}^{2}\Phi (s,\text{ }x)\text{ }+\text{ }\partial _{s}^{2}V(x)\Phi (s,\text{ }x)=0,NEWLINE\]NEWLINE for an analytic function \(\Phi (s,\text{{ }}x)\) with a discrete set of singularities. It turns out that the terms of the hyperasymptotic expansions for (1) can be recovered from studying the singularities of \(\Phi (s,\text{{ }}x)\) see \textit{O. Costin} and \textit{S. Garoufalidis} [Ann. Inst. Fourier 58, No. 3, 893--914 (2008; Zbl 1166.34055)]. The author re-examines Shatalov-Sternin's proof of the existence of resurgent solutions for (1), focusing the attention on the study of the structure of the Riemann surface of \(\Phi \).