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arXiv · 1807.09653

Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients

Ahmed Ghatasheh·Rudi Weikard
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Abstract

We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Also, we do not pose the definiteness condition customarily required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval $(a,b)$ are regular as well as the case of a $2\times2$ system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.

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BibTeXRIS

Ahmed Ghatasheh, Rudi Weikard. 2018-07-25. Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients. https://doi.org/10.1016/j.jde.2019.09.042

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