Versal unfolding of irregular singularities of a linear differential equation on the Riemann sphere
The author investigates the possibility to realize a linear differential operator \(P\) with unramified irregular singular points on the Riemann sphere as a confluence of singularities of a Fuchsian linear differential operator (called an unfolding of \(P\)) having the same index of rigidity. It is conjectured that this is always possible. There are given self-contained proofs of some known auxiliary results, such as the decomposition theorem for \(P\) or theorem describing basic formal solutions of the equation \(Pu=0\). Also, formal characteristic exponents of \(P\) at its singular point are defined and an analytic proof of the Fuchs-Hukuhara relation for their sum over all singular points is given, as an alternative to an algebraic approach of \textit{D. Bertrand} and \textit{G. Laumon} [``Appendice à exposants, vecteurs cycliques et majorations de multiplicités'', C. R. Conf. Franco-Japonaise, Preprint IRMA (1985)]. Then the definitions of generalized characteristic exponents (of generalized Riemann scheme) of \(P\) and of the spectral type of \(P\) are given. The latter determines the index of rigidity of \(P\). The conjecture is checked to be affirmative under some assumptions on the index of rigidity and spectral type of \(P\). Several examples of unfolding are presented.
scientific article; zbMATH DE number 7445180
scientific article