Singular Weyl-Titchmarsh-Kodaira theory for Jacobi operators (Q2869937)

The classical Weyl-Titchmarsh-Kodaira theory was originally developed for one-dimensional Schrödinger operators with one regular endpoint. It turned out recently that many aspects of this theory still can be established at a singular endpoint.NEWLINENEWLINEThe purpose of the present paper is to extend singular Weyl-Titchmarsh-Kodaira theory to the case of Jacobi operators NEWLINE\[NEWLINE (\tau f)(n)=a(n)f(n+1)+a(n-1)f(n-1)+b(n)f(n), \qquad n\in\mathbb{Z}, NEWLINE\]NEWLINE with the coefficients \(a(n)>0\), \(b(n)\in\mathbb{R}\). Compared to the case of Schrödinger operators, there are some significant differences in the proof of the main results due, in particular, to the fact that two sequences of coefficients are to be determined. The main result of the paper is a local Borg-Marchenko theorem. Moreover, the authors show that, in the case of purely discrete spectra, the spectral measure uniquely determines the operator, and establish a general Hochstadt-Liebermann-type uniqueness result. Finally, the authors use the connection with de Branges spaces of entire functions in order to give another general criterion when the spectral measure uniquely determines the Jacobi operator.