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Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators

Damanik, David and Killip, Rowan and Simon, Barry (2004) Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators. International Mathematics Research Notices, 2004 (22). pp. 1087-1097. ISSN 1073-7928. http://resolver.caltech.edu/CaltechAUTHORS:20110920-104550878

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Abstract

We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator −d^2/dx^2+V(x) on L^2 (0,∞) with V ∈ L2(0,∞) and the boundary condition u(0) = 0. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1155/S1073792804132790 DOIUNSPECIFIED
http://imrn.oxfordjournals.org/content/2004/22/1087.abstract?sid=2dbab605-fcc2-42fc-bba7-6648df778202PublisherUNSPECIFIED
Additional Information:© 2004 Hindawi Publishing Corporation. Received September 11, 2003. Revision received December 3, 2003. Accepted January 29, 2004. We would like to thank Roman Romanov for drawing our attention to [9]. David Damanik was supported in part by National Science Foundation (NSF) Grant DMS-0227289, and Barry Simon was supported in part by NSF Grant DMS-0140592.
Funders:
Funding AgencyGrant Number
NSFDMS-0227289
NSFDMS-0140592
Record Number:CaltechAUTHORS:20110920-104550878
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20110920-104550878
Official Citation:David Damanik, Rowan Killip, and Barry Simon Necessary and sufficient conditions in the spectral theory of Jacobi matrices and Schrödinger operators Int Math Res Notices (2004) Vol. 2004 1087-1097 doi:10.1155/S1073792804132790
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:25365
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:20 Sep 2011 20:05
Last Modified:20 Sep 2011 20:05

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