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

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

2005-03-18

arxiv.org/abs/math/0503392v1

Mathematics

Spectral Theory

Scientific paper

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) \mu_k^{-2n} + O(R^{-2n}) b_n &= \sum_{k=1}^{K(R)} p_k(n) \mu_k^{-2n+1} + O(R^{-2n}) $ where $1 < |\mu_j| < R$ for $j\leq K(R)$ 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.

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