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Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators

Damanik, David and Hundertmark, Dirk and Simon, Barry (2003) Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators. Journal of Functional Analysis, 205 (2). pp. 357-379. ISSN 0022-1236. doi:10.1016/S0022-1236(03)00070-3.

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For Jacobi matrices with a_n=1+(−1)^nαn−γ, b_n=(−1)^nβn−γ, we study bound states and the Szegő condition. We provide a new proof of Nevai's result that if γ>12, the Szegő condition holds, which works also if one replaces (−1)^n by cos(μn). We show that if α=0, β≠0, and γ<12, the Szegő condition fails. We also show that if γ=1, α and β are small enough (β^2+8α^2<1/24 will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).

Item Type:Article
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URLURL TypeDescription Paper
Damanik, David0000-0001-5924-3849
Hundertmark, Dirk0000-0002-0643-0138
Simon, Barry0000-0003-2561-8539
Additional Information:© 2003 Elsevier Inc. Received 22 August 2002, Accepted 2 January 2003, Available online 29 April 2003. Communicated by L. Gross We thank Rowan Killip, Paul Nevai, Mihai Stoiciu, and Andrej Zlatoš for valuable communications. Supported in part by NSF Grant DMS-0227289. Supported in part by NSF Grants DMS-9707661 and DMS-0140592.
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Issue or Number:2
Record Number:CaltechAUTHORS:20170602-143616540
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Official Citation:David Damanik, Dirk Hundertmark, Barry Simon, Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators, Journal of Functional Analysis, Volume 205, Issue 2, 2003, Pages 357-379, ISSN 0022-1236, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77926
Deposited By: Ruth Sustaita
Deposited On:03 Jun 2017 01:33
Last Modified:15 Nov 2021 17:35

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