Triangularisation of Singularly Perturbed Logarithmic Differential Systems of Rank 2

TL;DR

This paper develops a method to transform singularly perturbed linear differential systems of rank two into an upper-triangular form using holomorphic gauge transformations, facilitating the analysis of their asymptotic behavior.

Contribution

It introduces a Gevrey-regular gauge transformation approach to triangularize such systems and constructs a family of Levelt filtrations that interpolate between eigen-decompositions at different limits.

Findings

Systems can be upper-triangularized via Gevrey gauge transformations.

Constructed a family of Levelt filtrations parametrized by  that interpolate eigen-decompositions.

Revealed asymptotic eigen-decomposition behavior as   0 and as x  0.

Abstract

We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $\hbar x\partial_x \psi (x, \hbar) + A (x, \hbar) \psi (x, \hbar) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $\hbar$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $\hbar$ in the same sector. We use this result to construct a family in $\hbar$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $\hbar$; this family of filtrations recovers in the $\hbar \to 0$ limit the eigen-decomposition for the $\hbar$-leading-order of the matrix $A (x, \hbar)$, and also recovers in the $x \to 0$ limit the eigen-decomposition of the residue matrix $A (0, \hbar)$.