Spectral properties of unbounded Jacobi matrices with almost monotonic weights

TL;DR

This paper develops a unified framework to analyze the spectral properties of unbounded Jacobi matrices with almost monotonic weights, addressing several longstanding conjectures in orthogonal polynomials and stochastic processes.

Contribution

It introduces a novel approach to identify spectra of Jacobi matrices, applying it to resolve multiple open conjectures in related mathematical fields.

Findings

Confirmed spectral properties for specific classes of Jacobi matrices.

Provided new insights into the spectra of generators of birth and death processes.

Linked spectral analysis to longstanding conjectures in orthogonal polynomials.

Abstract

We present an unified framework to identify spectra of Jacobi matrices. We give applications to long-standing conjecture of Chihara concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent concerning continuous spectra of generators of birth and death processes and to spectral properties of operators studied by Janas, Moszy\'nski and Pedersen.