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Published December 20, 2003 | Submitted
Journal Article Open

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

Abstract

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).

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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August 19, 2023
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