On Fourier expansions for systems of ordinary differential equations with distributional coefficients

Abstract

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations 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. Specifically, we construct a generalized Weyl-Titchmarsh $m$-function with corresponding spectral measure $\tau$ and a generalized Fourier transform after imposing certain conditions on $J$, $q$, and $w$. Different conditions are motivated and studied in the later sections. A Fatou-type identity needed for our result is recorded in the appendix.