American Institute of Mathematical Sciences

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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

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), 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.

show all references

References:
[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.

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