On The Summability Of Formal Solutions In Liouville-green Theory
Journal Of Dynamical And Control Systems
Mathematics and Computer Science
We consider the second-order differential equation ∈2 y″ = (1+∈2ψ(x, ∈))y with a small parameter ∈, where ψ is even with respect to ∈. It is well known that it has two formal solutions y ±(x, ∈) = e ±x/∈ h±(x, ∈), where h ±(x, ∈) is a formal series in powers of ∈ whose coefficients are functions of x. It has been shown  that one resp. both of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of ψ. The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to ∈.
Bodine, Sigrun, and R. Schäfke. 2002. "On the summability of formal solutions in Liouville-Green theory." Journal Of Dynamical And Control Systems 8(3): 371-398.