The Jacobi Operator on $$(-1,1)$$ and Its Various m-Functions

IF 0.7 4区 数学 Q2 MATHEMATICS
Fritz Gesztesy, Lance L. Littlejohn, Mateusz Piorkowski, Jonathan Stanfill
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引用次数: 0

Abstract

We offer a detailed treatment of spectral and Weyl–Titchmarsh–Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression

$$\begin{aligned} \tau _{\alpha ,\beta } =&- (1-x)^{-\alpha } (1+x)^{-\beta }(d/dx) \big ((1-x)^{\alpha +1}(1+x)^{\beta +1}\big ) (d/dx), \\&\alpha , \beta \in {\mathbb {R}}, \, x \in (-1,1), \end{aligned}$$

in \(L^2\big ((-1,1); (1-x)^{\alpha } (1+x)^{\beta } dx\big )\), \(\alpha , \beta \in {\mathbb {R}}\). In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general \(\eta \)-periodic and Krein–von Neumann extensions. In particular, we treat all underlying Weyl–Titchmarsh–Kodaira and Green’s function induced m-functions and revisit their Nevanlinna–Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.

$$(-1,1)$$上的雅可比算子及其各种 m 函数
我们对微分表达式 $$\begin{aligned} 的所有自共雅各比算子实现的光谱和 Weyl-Titchmarsh-Kodaira 理论进行了详细论述。=&- (1-x)^{-\alpha }(1+x)^{-\beta }(d/dx) \big ((1-x)^{\alpha +1}(1+x)^{\beta +1}\big ) (d/dx), \&\alpha , \beta \in {\mathbb {R}}, \, x \in (-1,1), \end{aligned}$$in \(L^2\big ((-1,1); (1-x)^{\alpha }).(1+x)^{beta } dx\big )\),\(\alpha , \beta \in {\mathbb {R}}\).除了详细讨论导致雅可比正交多项式作为特征函数的分离边界条件外,我们还详尽地处理了耦合边界条件的情况,并借助一般的 \(\eta \)-periodic 和 Krein-von Neumann 扩展来说明后者。特别是,我们处理了所有底层的韦尔-蒂奇马什-柯达伊拉和格林函数诱导的 m 函数,并重温了它们的内万林纳-赫格洛茨性质。我们还考虑了与正交多项式相关的其他微分算子的联系,如 Laguerre、Gegenbauer 和 Chebyshev。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.20
自引率
12.50%
发文量
107
审稿时长
3 months
期刊介绍: Complex Analysis and Operator Theory (CAOT) is devoted to the publication of current research developments in the closely related fields of complex analysis and operator theory as well as in applications to system theory, harmonic analysis, probability, statistics, learning theory, mathematical physics and other related fields. Articles using the theory of reproducing kernel spaces are in particular welcomed.
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