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Finite Gap Jacobi Matrices, III. Beyond the Szegő Class (1108.0183v1)

Published 31 Jul 2011 in math.SP, math-ph, and math.MP

Abstract: Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $\omega_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $\omega_1,..., \omega_\ell$. Let ${\tilde{a}n, \tilde{b}_n}{n=1}\infty$ be a point in the isospectral torus for $\fre$ and $\tilde{p}n$ its orthogonal polynomials. Let ${a_n,b_n}{n=1}\infty$ be a half-line Jacobi matrix with $a_n = \tilde{a}n + \delta a_n$, $b_n = \tilde{b}_n + \delta b_n$. Suppose [ \sum{n=1}\infty %(\abs{a_n-\tilde{a}n}2 + \abs{b_n-\tilde{b}_n}2) <\infty \abs{\delta a_n}2 + \abs{\delta b_n}2 <\infty ] and $\sum{n=1}N e{2\pi i\omega n} \delta a_n$, $\sum_{n=1}N e{2\pi i\omega n} \delta b_n$ have finite limits as $N\to\infty$ for all $\omega$ in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to $\omega$, then for $z\in\bbC\setminus\bbR$, $p_n(z)/\tilde{p}_n(z)$ has a limit as $n\to\infty$. Moreover, we show that there are non-Szeg\H{o} class $J$'s for which this holds.

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