The Jacobi operator and its Donoghue $m$-functions

Abstract

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2((-1,1);(1-x{)}^{\alpha }(1+x{)}^{\beta }dx)$, associated to the differential expression \begin{align*} \begin{split} τ_{α,β} = - (1-x)^{-α} (1+x)^{-β}(d/dx) \big((1-x)^{α+ 1}(1+x)^{β+ 1}\big) (d/dx),& \\ x \in (-1,1), \; α, β\in \mathbb{R}, \end{split} \end{align*} whenever at least one endpoint, $x=\pm 1$, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's $m$-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case, a topic not covered in the literature.

28 pages. arXiv admin note: substantial text overlap with arXiv:2107.09832; text overlap with arXiv:2102.00685, arXiv:1910.13117

References in corpus (3)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators