Geodesic
Meromorphic Jost Functions and Asymptotic Expansions for Jacobi Parameters
Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 2, pp. 78-92

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We show that the parameters $a_n$, $b_n$ of a Jacobi matrix have a complete asymptotic expansion $$ a_n^2-1=\sum_{k=1}^{K(R)} p_k(n) \mu_k^{-2n}+ O(R^{-2n}),\qquad b_n=\sum_{k=1}^{K(R)} p_k(n)\mu_k^{-2n+1}+O(R^{-2n}), $$ where $1|\mu_j|$ 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.
Keywords: Jost function, exponential decay.
Mots-clés : Jacobi matrix 
@article{FAA_2007_41_2_a5,
     author = {B. Simon},
     title = {Meromorphic {Jost} {Functions} and {Asymptotic} {Expansions} for {Jacobi} {Parameters}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {78--92},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2007_41_2_a5/}
}
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B. Simon. Meromorphic Jost Functions and Asymptotic Expansions for Jacobi Parameters. Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 2, pp. 78-92. http://geodesic.mathdoc.fr/item/FAA_2007_41_2_a5/