Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

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 \[ \sum_{n=1}^\infty %(\abs{a_n-\tilde{a}_n}^2 + \abs{b_n-\tilde{b}_n}^2) <\infty \abs{δa_n}^2 + \abs{δb_n}^2 <\infty \] 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 $z\in \mathbb{C}\setminus \mathbb{R}$, $p_n(z)/{\tilde{p}}_n(z)$ has a limit as $n\to \infty$. Moreover, we show that there are non-Szegő class $J$'s for which this holds.

15 pages