Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem

TL;DR

This paper proves a uniqueness result for reflectionless Jacobi matrices with a specific parameter and discusses its generalizations, leading to applications that relax assumptions in the Denisov-Rakhmanov Theorem.

Contribution

It establishes a new uniqueness theorem for reflectionless Jacobi matrices and extends the Denisov-Rakhmanov Theorem by relaxing certain conditions.

Findings

Reflectionless Jacobi matrices with a single parameter equal to 1 are uniquely the free matrix.

Generalizations to arbitrary sets are discussed.

Applications include dropping assumptions in the Denisov-Rakhmanov Theorem when some parameters are close to 1.

Abstract

If a Jacobi matrix $J$ is reflectionless on $(-2,2)$ and has a single $a_{n_0}$ equal to 1, then $J$ is the free Jacobi matrix $a_n\equiv 1$, $b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary sets and present several applications, including the following: if a Jacobi matrix has some portion of its $a_n$'s close to 1, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.