On the summability of formal solutions in Liouville-Green theory
The authors consider the second-order linear differential equation \[ \varepsilon ^2y''=(1+\varepsilon ^2\psi (x,\varepsilon))y\tag{*} \] where \(\varepsilon\) is a small complex parameter and \(\psi\) is complex-analytic w.r.t. \((x,\varepsilon)\) and even w.r.t. \(\varepsilon\). It is known [see \textit{F. W. J. Olver}, Asymptotics and special functions. New York-London: Academic Press (1974; Zbl 0303.41035), Chapter 10], that equation \((*)\) has two formal solutions \(y^{\pm x/\varepsilon }h^{\pm}(x,\varepsilon)\), where \(h^{\pm}(x,\varepsilon)\) is a formal power series of \(\varepsilon\) whose coefficients are functions analytic in \(x\). It has been shown in \textit{T. M. Dunster, D. A. Lutz} and \textit{R. Schäfke} [Proc. R. Soc. Lond., Ser. A 440, 37-54 (1993; Zbl 0798.34083)] that one or both of these solutions are \(1\)-summable in certain directions if the \(x\)-domain of \(\psi\) satisfies certain conditions. The authors of the present paper give necessary (and sufficient) conditions for the \(1\)-summability of one or both of the above formal solutions in terms of \(\psi\). The method of proof relies on the construction of a differential equation of the kind of \((*)\), with \(\psi\) of the form \(\varepsilon ^2\varphi (x)\), which exhibits a prescribed Stokes phenomenon w.r.t. \(\varepsilon\).
scientific article; zbMATH DE number 1845282