Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: A Levinson type theorem on complex domains, and applications

Abstract

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 $\left|z\right|$. Inspired by [60], we introduce two conditions on the dominant diagonal term (the L-condition) and on the perturbation term (the good decay condition) of the coefficients of the system, respectively. Assuming the validity of these conditions, we then show the existence and uniqueness, on big sectorial domains, of an asymptotic fundamental matrix solution, i.e. asymptotically equivalent (for large $\left|z\right|$) 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, the leading diagonal term of the coefficient being a generalized polynomial in z with real exponents. 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/IM correspondence). Notably, our results fill two significant gaps in the mathematical literature pertaining to these conjectural relations.

Finally, as a second application, we consider the classical case of ODEs with meromorphic coefficients. Under an adequateness condition on the coefficients (allowing ramification of the irregular singularities), we show that our results reproduce (with a shorter proof) the main asymptotic existence theorems of Y. Sibuya [80], [81] and W. Wasow [94] in their optimal refinements: the sectors of validity of the asymptotics are maximal, and the asymptotic fundamental system of solutions is unique.