Article Contents
Article Contents

# Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator

• In this paper we give a first order system of difference equations which provides a useful companion system in the study of Jacobi matrix operators and make use of it to obtain a characterization of the spectral density function for a simple case involving absolutely continuous spectrum on the stability intervals.
Mathematics Subject Classification: Primary: 39A70, 39A23; Secondary: 47B36, 47B39, 47A75.

 Citation:

•  [1] M. Appell, Sur la transformation des équations différentielles linéaires, Comptes rendus hebdomadaires des seánces de l'Académie des sciences 91 (4) (1880), 211-214. [2] F.V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, N.Y., 1964. [3] M.S.P. Eastham, "The Spectral Theory of periodic differential equations," Scottish Academic Press, London, 1973. [4] C.T. Fulton, D.B. Pearson, and S. Pruess, New characterizations of spectral density functions for singular Sturm-Liouville problems, J. Comput. Appl. Math (2008) 212 (2), pp. 194-213. [5] C.T. Fulton, D.B. Pearson, and S. Pruess, Efficient calculation of spectral density functions for specific classes of singular Sturm-Liouville problems, J. Comput. Appl. Math (2008) 212 (2), pp. 150-178. [6] C.T. Fulton, D.B. Pearson, and S. Pruess, Algorithms for Estimating Spectral Density Functions for Periodic Potentials, preprint, arXiv:1303.5878. [7] C.T. Fulton, D.B. Pearson, and S. Pruess, Titchmarsh-Weyl theory for tridiagonal Jacobi matrices and computation of their spectral functions, in "Advances in nonlinear analysis: theory, methods and applications," (ed. S. Sivasundaram), Math Probl. Eng. Aerosp. Sci., 3, Camb. Sci. Publ.,(2009), 165-172. [8] B. Simon, "Szegö's Theorem and Its Descendants," Princeton University Press, Princeton, 2011. [9] G. Stolz and R. Weikard, "Notes of Seminar on Jacobi Matrices," Dept of Mathematics, University of Alabama, Birmingham, Jan. 2004. [10] G. Teschl, Jacobi Operators and Completely, Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, Vol 72, Amer. Math. Soc., 2000.
Open Access Under a Creative Commons license