Jost functions and Jost solutions for Jacobi matrices, III. Asymptotic series for decay and meromorphicity

Abstract

We show that the parameters $a_n,b_n$ of a Jacobi matrix have a complete asymptotic series $ a_n^2 -1 &= \sum_{k=1}^{K(R)} p_k(n) μ_k^{-2n} + O(R^{-2n}) b_n &= \sum_{k=1}^{K(R)} p_k(n) μ_k^{-2n+1} + O(R^{-2n}) $ where $1<|{\mu }_j|<R$ for $j\le K(R)$ and all $R$ if and only if the Jost function, $u$, written in terms of $z$ (where $E=z+z^{-1}$) is an entire meromorphic function. We relate the poles of $u$ to the ${\mu }_j$'s.