Finite Gap Jacobi Matrices, III. Beyond the Szegő Class

Abstract

Let ${\frak {e}}\subset {\mathbb {R}}$ be a finite union of ℓ+1 disjoint closed intervals, and denote by ω _j the harmonic measure of the j left-most bands. The frequency module for ${\frak {e}}$ is the set of all integral combinations of ω ₁,…,ω _ℓ . Let $\{\tilde{a}_{n}, \tilde{b}_{n}\}_{n=-\infty}^{\infty}$ be a point in the isospectral torus for ${\frak {e}}$ 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

$$\sum_{n=1}^\infty \lvert \delta a_n\rvert ^2 + \lvert \delta b_n\rvert ^2 <\infty $$

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→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, $p_{n}(z)/\tilde{p}_{n}(z)$ has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds.