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Finite gap Jacobi matrices. III: Beyond the Szegő class. (English) Zbl 1238.42009
Summary:
[Parts I and II in ibid. 32, No. 1, 1–65 (2010; Zbl 1200.42012) and ibid. 33, No. 3, 365–403 (2011; Zbl 1236.42021)]
Let \({\mathfrak{e}} \subset {\mathbb{R}}\) be a finite union of \(\ell+1\) disjoint closed intervals, and denote by \(\omega_{j}\) the harmonic measure of the \(j\) left-most bands. The frequency module for \({\mathfrak{e}}\) is the set of all integral combinations of \(\omega_{1},\dots,\omega_{\ell}\). Let \(\{{\tilde{a}}_{n},{\tilde{b}}_{n}\}_{n=-\infty}^{\infty}\) be a point in the isospectral torus for \({\mathfrak{e}}\) 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 \[ \sum_{n=1}^\infty | \delta a_n| ^2 + | \delta b_n| ^2 <\infty \] 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 \(z \in \mathbb{C} \setminus \mathbb{R}\), \(p_{n}(z)/{\tilde{p}}_{n}(z)\) has a limit as \(n \to \infty\). Moreover, we show that there are non-Szegő class \(J\)’s for which this holds.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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References:
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