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Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights

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Abstract

We present two approaches to the spectral studies for infinite Jacobi matrices with monotonic or “near-to-monotonic” weights. The first one is based on the subordination theory due to Khan and Pearson [17] combined with the detailed analysis of the transfer matrices for the solutions of the formal eigenequation. The second one uses an extension of the commutator approach developed by Putnam in [19]. Applying these methods we prove the absolute continuity for several classes of weights and diagonals. For some other cases we prove the emptiness of the point spectrum. The results are illustrated with examples and compared with the results of Dombrowski [7]-[13], Clark [2] and of Máté and Nevai [18]. We show that some of our results are stronger.

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Authors and Affiliations

  1. Oddział w Krakowie, Instytut Matematyczny Polskiej Akademii Nauk, ul. Św. Tomasza 30, 31-027, Kraków

    Jan Janas

  2. Wydzial Matematyki Informatyki i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097, Warszawa

    Marcin Moszyński

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  1. Jan Janas
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  2. Marcin Moszyński
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The research of the first author has been supported by the KBN grant PB 2 P03A 002 13.

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Janas, J., Moszyński, M. Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights. Integr equ oper theory 43, 397–416 (2002). https://doi.org/10.1007/BF01212702

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