The Jacobi operator on $(-1,1)$ and its various $m$-functions

TL;DR

This paper provides a comprehensive analysis of the spectral theory and m-functions for all self-adjoint realizations of a class of Jacobi operators on (-1,1), including boundary conditions, extensions, and connections to orthogonal polynomials.

Contribution

It offers a detailed, unified treatment of spectral and Weyl-Titchmarsh theory for Jacobi operators with various boundary conditions, including new insights into m-functions and extensions.

Findings

Exhaustive characterization of boundary conditions leading to Jacobi polynomials.

Analysis of all associated m-functions and their properties.

Connections established with other orthogonal polynomial operators.

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{align*} \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{align*} 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.