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Published 2004 | public
Journal Article

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

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.

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.

Additional details

Created:
August 19, 2023
Modified:
October 24, 2023