The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl $m$-function for several self-adjoint extensions with interesting boundary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna--Herglotz $m$-functions are, to the best knowledge of the author, the first explicit examples to stem from singular higher-order differential equations. The creation of the boundary triples involves taking pieces, determined in a previous paper, of the principal and non-principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space. These maps act like quasi-derivatives, which are usually not well-defined for all functions in the maximal domain of singular expressions. However, well-defined regularizations of quasi-derivatives are produced by putting the pieces of the non-principal solutions through a modified Gram--Schmidt process.

中文翻译：

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雅可比微分算子幂的边界三元组和外尔 m 函数

边界三元组的抽象理论被应用于经典的雅可比微分算子及其幂，以获得具有有趣边界条件的几个自伴随扩展的 Weyl $m$ 函数：分离的、周期性的和产生弗里德里希扩展的那些扩展。据作者所知，这些矩阵值 Nevanlinna--Herglotz $m$-函数是第一个源自奇异高阶微分方程的显式示例。边界三元组的创建涉及将在前一篇论文中确定的微分方程的主解和非主解的碎片放入倍半线性形式，以产生从最大域到边界空间的映射。这些映射就像准导数 对于奇异表达式的最大域中的所有函数，它们通常没有明确定义。然而，通过将非主解的片段放入经过修改的 Gram--Schmidt 过程，可以产生准导数的明确正则化。