Finite Gap Jacobi Matrices, III. Beyond the Szeg\H{o} Class

Let $\text{fre}\subset \text{bbR}$ be a finite union of $\mathrm{\ell }+1$ disjoint closed intervals and denote by ${\omega }_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\text{fre}$ is the set of all integral combinations of ${\omega }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\omega }_{\mathrm{\ell }}$. Let $\{{\tilde{a}}_n,{\tilde{b}}_n{\}}_{n=1}^{\mathrm{\infty }}$ be a point in the isospectral torus for $\text{fre}$ and ${\tilde{p}}_n$ its orthogonal polynomials. Let $\{a_n,b_n{\}}_{n=1}^{\mathrm{\infty }}$ be a half-line Jacobi matrix with $a_n={\tilde{a}}_n+\delta a_n$, $b_n={\tilde{b}}_n+\delta b_n$. Suppose $$\sum _{n=1}^{\mathrm{\infty }}\text{abs}{\delta a_n}^2+\text{abs}{\delta b_n}^2<\mathrm{\infty }$$ 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 $N\to \mathrm{\infty }$ for all $\omega$ in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to $\omega$, then for $z\in \text{bbC}\setminus \text{bbR}$, $p_n(z)\mathrm{/}{\tilde{p}}_n(z)$ has a limit as $n\to \mathrm{\infty }$. Moreover, we show that there are non-Szeg\H{o} class $J$’s for which this holds.