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The absolutely continuous spectrum of Jacobi matrices. (English) Zbl 1235.47032
Let \(H\) be a one-dimensional discrete Schrödinger operator on \({\ell_2}\) \((Hu)(n)=u(n+1)+u(n-1)+V(n)u(n)\) with a bounded half line potential \(V\). Let \(S\) be the shift in \({\ell_2}\), and let \(\omega(V)\) be the set of right limits of \(S^kV\) in the distance \(d(V,W)=\sum 2^{-|n|}|V(n)-W(n)|\).
The main result of the paper is the following: Every bounded whole line potential \(W\in \omega(V)\) is reflectionless on the essential support \(\Sigma_{ac}\) of the absolutely continuous part of the spectral measure of \(V\). Recall that a bounded potential \(W\) is said to be reflectionless on a Borel set \(A(\subset {\mathbb R})\) if \(m_+(t)=-\overline{m_-(t)}\) for a.e. \(t\in A\), \(m_\pm\) being the Titchmarsh-Weyl \(m\)-functions of the operator \(H\) restricted to the half lines \({\mathbb Z}_\pm\). This statement is also proved for a class of bounded Jacobi matrices. The proof is essentially based on the asymptotic value distribution formula from [S. V. Breimesser and D. B. Pearson, Math. Phys. Anal. Geom. 3, No. 4, 385–403 (2000; Zbl 1016.47033)].
An important consequence of the main theorem of the present paper is the Oracle Theorem which allows to predict future values of potentials with absolutely continuous spectrum, based on information on past values. As another corollary, it is shown that for the half-line restrictions \(W_\pm\) of \(W\in\omega(V)\) to \({\mathbb Z}_\pm\), the following inclusions hold: \(\Sigma_{ac}(W_\pm)\supset\Sigma_{ac}(V)\), a result which is due to [Y. Last and B. Simon, Invent. Math. 135, No. 2, 329–367 (1999; Zbl 0931.34066)]. One more consequence of this theorem is a Denisov-Rakhmanov type result for finite gap Jacobi coefficients.

MSC:
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47A10 Spectrum, resolvent
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