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 τ 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.