Polynomial birth--death processes and the second conjecture of Valent

TL;DR

This paper proves Valent's conjecture regarding the classification of Jacobi matrices with weights that grow polynomially, advancing understanding in spectral theory.

Contribution

It provides a proof of Valent's conjecture on the spectral type of certain Jacobi matrices with polynomial weights.

Findings

Valent's conjecture is confirmed.

Jacobi matrices with polynomially growing weights are classified.

The spectral type of these matrices is determined.

Abstract

The conjecture of Valent about the type of Jacobi matrices with polynomially growing weights is proved.