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

2023/03/03 by Steven Redolfi, Redolfi, Steven, Rudi Weikard +1 · 1 citation
Mathematics · #34L10 #47A70 #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2303.02086

openalex publication_date 2023/03/03 · openalex created_date 2023/03/07 · openalex updated_date 2026/07/28

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.

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