arXiv 1901.062712019-01math.CA

Jacobi微分算子幂域的性质与分解

Properties and Decompositions of Domains for Powers of the Jacobi Differential Operator

Dale Frymark, Constanze Liaw

发表信息：: J. Math. Anal. Appl., vol. 489, iss. 1 (2020) 124-155
DOI：: 10.1016/j.jmaa.2020.124155

摘要：

我们着手建立一个框架的Jacobi微分算子的权力，不使用经典的缺乏元素的自伴扩展理论。相反，我们依赖于更简单的功能，捕捉这些元素对扩展的影响，但由边界渐近定义。这种新的视角使计算变得更加容易，并允许对相关领域进行更细致的分析。Jacobi算子的$n$ -th复合的最大域用每个导数的光滑条件来表征，并且对于$j\in\mathbb{N}_0$，底层希尔伯特空间中的函数的端点行为可以被分类为$(1-x)^j$、$(1+x)^j$、$(1-x)^{-α+j}$和$(1+x)^{β+j}$。大多数这些行为只能发生在函数在相关的最小域，这导致了一个公式的缺陷空间与一个方便的基础。然后，利用GKN理论，以亏空间的新基函数给出了自伴扩张，包括重要的左定域。对Laguerre算子也作了评述。

英文摘要：

We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains. The maximal domain for $n$-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of functions in the underlying Hilbert space can then be classified, for $j\in\mathbb{N}_0$, by $(1-x)^j$, $(1+x)^j$, $(1-x)^{-α+j}$ and $(1+x)^{β+j}$. Most of these behaviors can only occur when functions are in the associated minimal domain, and this leads to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are then given in terms of the new basis functions for the defect spaces using GKN theory. Comments are made for the Laguerre operator as well.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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