Liouville transforms for ordinary differential equations with distributional coefficients

Abstract

We study Liouville transforms for the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order $0$ with $q$ Hermitian and $w$ non-negative. In particular, we examine their use in the treatment of quasi-regular endpoints and their impact on the Weyl-Titchmarsh $\mathbb M$-function associated with the differential equation.