In this paper, we consider systems of linear ordinary differential equations,
with analytic coefficients on big sectorial domains, which are asymptotically
diagonal for large values of β£zβ£. Inspired by N. Levinson's work [Lev48], we
introduce two conditions on the dominant diagonal term (the L-condition)
and on the perturbation term (the gooddecaycondition) of the
coefficients of the system, respectively. Under these conditions, we show the
existence and uniqueness, on big sectorial domains, of an asymptotic
fundamental matrix solution, i.e. asymptotically equivalent (for large β£zβ£)
to a fundamental system of solutions of the unperturbed diagonal system.
Moreover, a refinement (in the case of subdominant solutions) and a
generalization (in the case of systems depending on parameters) of this result
are given.
As a first application, we address the study of a class of ODEs with
not-necessarily meromorphic coefficients. We provide sufficient conditions on
the coefficients ensuring the existence and uniqueness of an asymptotic
fundamental system of solutions, and we give an explicit description of the
maximal sectors of validity for such an asymptotics. Furthermore, we also focus
on distinguished examples in this class of ODEs arising in the context of open
conjectures in Mathematical Physics relating Integrable Quantum Field Theories
and affine opers (ODE/IMcorrespondence). Our results fill two significant
gaps in the mathematical literature pertaining to these conjectural relations.
As a second application, we consider the classical case of ODEs with
meromorphic coefficients. Under an adequateness condition on the
coefficients, we show that our results reproduce (with a shorter proof) the
main asymptotic existence theorems of Y. Sibuya [Sib62, Sib68] and W. Wasow
[Was65] in their optimal refinements.Comment: 43 pages, 7 figure