Finite Gap Jacobi Matrices, III. Beyond the Szegő Class
Jacob S. Christiansen, Barry Simon, Maxim Zinchenko · arXiv (Cornell University) · 2011
Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $ω_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $ω_1,..., ω_\ell$. 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 $a_n = \tilde{a}_n + δa_n$, $b_n = \tilde{b}_n + δb_n$. Suppose \[ \sum_{n=1}^\infty %(\abs{a_n-\tilde{a}_n}^2 + \abs{b_n-\tilde{b}_n}^2)