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Probabilistic Averages of Jacobi Operators

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Abstract

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ρ (mod 1)) for ρ > 0 not an integer.

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  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Helge Krüger

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  1. Helge Krüger
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Correspondence to Helge Krüger.

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Communicated by B. Simon

H. K. was supported by NSF grant DMS–0800100 and a Nettie S. Autrey Fellowship.

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Krüger, H. Probabilistic Averages of Jacobi Operators. Commun. Math. Phys. 295, 853–875 (2010). https://doi.org/10.1007/s00220-010-1014-y

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