In this work, we consider the n -dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

中文翻译：

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Sturm-Liouville问题的高维分数分析

在这项工作中，我们通过使用涉及左侧和右侧黎曼-利维尔分数阶导数的梯度算子的分数形式来考虑n维分数Sturm-Liouville特征值问题。我们研究了本征函数的主要性质以及相关分数阶边界问题的本征值。更准确地说，我们证明了特征函数是正交的，特征值是真实而简单的。此外，使用分数阶变分法的技术，我们在主要结果中证明了特征值是分离的并形成无限序列，其中特征值可以根据幅度的增加进行排序。最后，建立了与Clifford分析的联系。