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

TL;DR

The paper develops a Levinson-type theorem on complex domains for linear systems $\frac{dY}{dz}=(\Lambda(z)+R(z))Y$, introducing the $L$-condition on the dominant diagonal term and the $good\ decay$ condition on the perturbation. It proves existence and uniqueness of an asymptotic fundamental matrix $Y$ satisfying $Y(z)=(I_n+o(1))\exp(\int_{z_0}^z\Lambda(w)\,dw)$ as $|z|\to\infty$ in wide sectors, and extends the results to subdominant solutions and parametric families with uniform control. The theory applies to ODEs with not-necessarily meromorphic coefficients, yielding maximal sectors of validity and enabling applications to the ODE/IM correspondence, while also recovering classical meromorphic-coefficient results (Sibuya–Wasow) in a shorter, unified framework. Overall, it provides a robust, general method for constructing and uniquely characterizing asymptotic fundamental systems in complex domains, with significant implications for integrable systems and mathematical physics.

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 $|z|$. 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 $good\,\,decay\,\,condition$) 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 $|z|$) 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 ($ODE/IM\,\,correspondence$). 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.

Figures (7)

• Figure 1.1: The domain $U$, union of lines (some are represented) with natural orientation. Also an example of an oriented segment $[z_1,z_2]$ is represented.
• Figure 2.1: The half plane ${\mathbb H}_{\varphi,a}$. The angles are represented mod $2\pi$. Equivalently, the figure may be intended as the representation of the projection onto ${\mathbb C}$ of the sheet of $\widetilde{{\mathbb C}^*}$ where ${\mathbb H}_{\varphi,a}$ lies.
• Figure 2.2: The domain ${\mathbb H}_{I,a}$. The angles are represented mod $2\pi$. Equivalently, the figure is the projection onto ${\mathbb C}$. Two lines $\ell_{\varphi,b}$ and $\ell_{\varphi^\prime,b^\prime}$ are also represented, with $\varphi_{\mathrm{min}}<\varphi<\varphi^\prime<\varphi_{\mathrm{max}}$
• Figure 2.3:
• Figure 2.4: $z$ goes to infinity along $\widehat{\ell}$.

Theorems & Definitions (84)

• Theorem 1.1: Theorem \ref{['mainth']}
• Definition 2.1
• Definition 2.2
• Example 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Corollary 2.7
• Lemma 2.8
• Lemma 2.9