Monotone Jacobi parameters and non-Szegő weights

Creators: Kreimer, Yury; Last, Yoram; Simon, Barry

Abstract

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n ≡ 1, b_n = −Cn^(−β)(0 < β , 2/3), one has dµ(x) = w(x) on (−2,2), and near x = 2, w(x) = e^(-2Q(x)) where Q(x) = β^(-1)C^(1/β) Г(3/2)Г(1/β-1/2)(2-x)1/2-1/β/Г(1/β +1) (1 + o((2-x)))).

Additional Information

© 2009 Elsevier B.V. Received 8 November 2007; received in revised form 4 April 2008; accepted 28 April 2008. Available online 9 November 2008. Communicated by Sergey Krushchev. The first author was supported in part by The Israel Science Foundation (grant no. 1169/06). The third author was supported in part by NSF grant DMS-0140592. The second and the third authors' research was supported in part by Grant No. 2002068 and No. 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. It is a pleasure to thank Fritz Gesztesy, Uri Kaluzhny, and Doron Lubinsky for useful discussions. The authors would also like to thank Mira Shamis and the referees for pointing out several corrections to the original manuscript. B.S. would like to thank Ehud de Shalit for the hospitality of the Einstein Institute of Mathematics at the Hebrew University where some of this work was done. Y.L. would like to thank Matthias Flach for the hospitality of the Department of Mathematics at Caltech where some of this work was done.

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