Department of Mathematics and Statistics, San Diego State University

Seminars on Differential and Difference Equations,

and Dynamical Systems

An inverse problem in Liouville-Green theory.

Sigrun Bodine, University of Puget Sound, Tacoma, WA.

We consider the second order differential equation tex2html_wrap_inline21 with a small parameter tex2html_wrap_inline23 where tex2html_wrap_inline25 is even with respect to tex2html_wrap_inline23 . It is known that it has two formal solutions tex2html_wrap_inline29 , where tex2html_wrap_inline31 is a formal series in powers of tex2html_wrap_inline23 whose coeffiecients are functions of x.

Dunster, Lutz and Schäfke showed in 1993 that one resp. both of these solutions are summable in certain directions if tex2html_wrap_inline25 satisfies certain conditions, in particular concerning its x-domain.

After reviewing these known results, we give necessary conditions for the summability of one or both of the above formal solutions in terms of tex2html_wrap_inline25 . 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 tex2html_wrap_inline23 .

This research is joint work with Reinhard Schäfke from the Université Louis Pasteur in Strasbourg, France.

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