Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem

Remling, Christian

doi:10.1090/S0002-9939-2010-10747-5

Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem
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by Christian Remling

Proc. Amer. Math. Soc. 139 (2011), 2175-2182

DOI: https://doi.org/10.1090/S0002-9939-2010-10747-5

Published electronically: November 30, 2010

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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 $a_n\equiv 1$, $b_n\equiv 0$. The paper discusses this result and its generalization to arbitrary sets and presents 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.

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