Abstract
The present paper reviews the theory of bounded Jacobi matrices whose essential spectrum is a finite gap set, and it explains how the theory can be extended to also cover a large number of infinite gap sets. Two of the central results are generalizations of Denisov–Rakhmanov’s theorem and Szegő’s theorem, including asymptotics of the associated orthogonal polynomials. When the essential spectrum is an interval, the natural limiting object \( J_0 \) has constant Jacobi parameters. As soon as gaps occur, \(\ell\) say, the complexity increases and the role of \( J_0 \) is taken over by an \(\ell\)-dimensional isospectral torus of periodic or almost periodic Jacobi matrices.
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