Resonant Confluence of Singular Points and Stokes Phenomena

Abstract

We consider a linear ordinary differential equation with resonant irregular singularity of generic type. For its generic deformation that splits the irregular singularity of the unperturbed equation into Fuchsian singularities of the perturbed one, the nonformal analytic classification invariants (Stokes operators) of the unperturbed equation are expressed via limit transition operators that compare appropriate monodromy eigenbases of the perturbed equation. We do this for all values of the Poincare rank and the dimension (denoted by k and n respectively), except for the case where k e 1 and n ≥ 3. We show (Theorems 2.1 and 2.2 in Sec. 2) that appropriate branches of the monodromy eigenfunctions of the perturbed equation converge to appropriate canonical solutions of the unperturbed equation. In the case where k e n e 2, this statement implies that the Stokes operators are limits of transition operators between appropriate monodromy eigenbases of the perturbed equation (Corollary 2.1). We give a generalization of the last statement for the case of higher Poincare rank and dimension (Corollary 2.2).