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.

中文翻译：

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具有分布系数的常微分方程组的谱理论

我们研究了实数区间 $(a,b)$ 上微分方程的一阶系统 $Ju'+qu=wf$ 的谱理论，当 $J$ 是常数可逆斜赫米特矩阵且 $q$和 $w$ 是矩阵，其条目是零阶分布，$q$ Hermitian 和 $w$ 非负。此外，我们没有提出方程系数通常需要的确定性条件。具体来说，我们构造最小和最大关系，并研究最大关系的自伴随限制。对于这些，我们确定格林函数并证明谱（或广义傅立叶）变换的存在。我们仔细研究了区间 $(a,b)$ 的端点是规则的特殊情况以及 $2\times2$ 系统的情况。