Triangularisation of Singularly Perturbed Logarithmic Differential Systems of Rank 2

Abstract

We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $\hslash x{\partial }_x\psi (x,\hslash )+A(x,\hslash )\psi (x,\hslash )=0$, with a regular singularity at $x=0$, and with a fixed asymptotic regularity in the perturbation parameter $\hslash$ 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 $\hslash$ in the same sector. We use this result to construct a family in $\hslash$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $\hslash$; this family of filtrations recovers in the $\hslash \to 0$ limit the eigen-decomposition for the $\hslash$-leading-order of the matrix $A(x,\hslash )$, and also recovers in the $x\to 0$ limit the eigen-decomposition of the residue matrix $A(0,\hslash )$.

arXiv comment17 pages; substantial changes from previous version, including the title