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

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2013, 2013(special): 247-257. doi: 10.3934/proc.2013.2013.247

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

Charles Fulton ^1, , David Pearson ^2, and Steven Pruess ^3,

1.	Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, 32901-6975, United States
2.	Department of Mathematics, University of Hull, Cottingham Road, Hull HU6 7RX, United Kingdom
3.	1133 N Desert Deer Pass, Green Valley, Arizona 85614-5530, United States

Received September 2012 Revised April 2013 Published November 2013

Abstract
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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.

Keywords: stability intervals., Periodic Jacobi matrix, absolutely continuous spectrum, discrete Schrödinger operator, Hill discriminant, Floquet solutions.

Mathematics Subject Classification: Primary: 39A70, 39A23; Secondary: 47B36, 47B39, 47A7.

Citation: Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

References:

[1]	M. Appell, Sur la transformation des équations différentielles linéaires,, Comptes rendus hebdomadaires des seánces de l'Académie des sciences 91 (4) (1880), 91 (1880), 211. Google Scholar
[2]	F.V. Atkinson, "Discrete and Continuous Boundary Problems,", Academic Press, (1964). Google Scholar
[3]	M.S.P. Eastham, "The Spectral Theory of periodic differential equations,", Scottish Academic Press, (1973). Google Scholar
[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), 212* (2008), 194. Google Scholar
[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), 212* (2008), 150. Google Scholar
[6]	C.T. Fulton, D.B. Pearson, and S. Pruess, Algorithms for Estimating Spectral Density Functions for Periodic Potentials, preprint,, , (). Google Scholar
[7]	C.T. Fulton, D.B. Pearson, and S. Pruess, Titchmarsh-Weyl theory for tridiagonal Jacobi matrices and computation of their spectral functions,, in, (2009), 165. Google Scholar
[8]	B. Simon, "Szegö's Theorem and Its Descendants,", Princeton University Press, (2011). Google Scholar
[9]	G. Stolz and R. Weikard, "Notes of Seminar on Jacobi Matrices,", Dept of Mathematics, (2004). Google Scholar
[10]	G. Teschl, Jacobi Operators and Completely, Integrable Nonlinear Lattices,, Mathematical Surveys and Monographs, (2000). Google Scholar

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References:

[1]	M. Appell, Sur la transformation des équations différentielles linéaires,, Comptes rendus hebdomadaires des seánces de l'Académie des sciences 91 (4) (1880), 91 (1880), 211. Google Scholar
[2]	F.V. Atkinson, "Discrete and Continuous Boundary Problems,", Academic Press, (1964). Google Scholar
[3]	M.S.P. Eastham, "The Spectral Theory of periodic differential equations,", Scottish Academic Press, (1973). Google Scholar
[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), 212* (2008), 194. Google Scholar
[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), 212* (2008), 150. Google Scholar
[6]	C.T. Fulton, D.B. Pearson, and S. Pruess, Algorithms for Estimating Spectral Density Functions for Periodic Potentials, preprint,, , (). Google Scholar
[7]	C.T. Fulton, D.B. Pearson, and S. Pruess, Titchmarsh-Weyl theory for tridiagonal Jacobi matrices and computation of their spectral functions,, in, (2009), 165. Google Scholar
[8]	B. Simon, "Szegö's Theorem and Its Descendants,", Princeton University Press, (2011). Google Scholar
[9]	G. Stolz and R. Weikard, "Notes of Seminar on Jacobi Matrices,", Dept of Mathematics, (2004). Google Scholar
[10]	G. Teschl, Jacobi Operators and Completely, Integrable Nonlinear Lattices,, Mathematical Surveys and Monographs, (2000). Google Scholar

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