American Institute of Mathematical Sciences

Advanced
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. Google Scholar

[2]

F.V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, N.Y., 1964.  Google Scholar

[3]

M.S.P. Eastham, "The Spectral Theory of periodic differential equations," Scottish Academic Press, London, 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), pp. 194-213.  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), pp. 150-178.  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 "Advances in nonlinear analysis: theory, methods and applications," (ed. S. Sivasundaram), Math Probl. Eng. Aerosp. Sci., 3, Camb. Sci. Publ.,(2009), 165-172.  Google Scholar

[8]

B. Simon, "Szegö's Theorem and Its Descendants," Princeton University Press, Princeton, 2011.  Google Scholar

[9]

G. Stolz and R. Weikard, "Notes of Seminar on Jacobi Matrices," Dept of Mathematics, University of Alabama, Birmingham, Jan. 2004. Google Scholar

[10]

G. Teschl, Jacobi Operators and Completely, Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, Vol 72, Amer. Math. Soc., 2000.  Google Scholar

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. Google Scholar

[2]

F.V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, N.Y., 1964.  Google Scholar

[3]

M.S.P. Eastham, "The Spectral Theory of periodic differential equations," Scottish Academic Press, London, 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), pp. 194-213.  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), pp. 150-178.  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 "Advances in nonlinear analysis: theory, methods and applications," (ed. S. Sivasundaram), Math Probl. Eng. Aerosp. Sci., 3, Camb. Sci. Publ.,(2009), 165-172.  Google Scholar

[8]

B. Simon, "Szegö's Theorem and Its Descendants," Princeton University Press, Princeton, 2011.  Google Scholar

[9]

G. Stolz and R. Weikard, "Notes of Seminar on Jacobi Matrices," Dept of Mathematics, University of Alabama, Birmingham, Jan. 2004. Google Scholar

[10]

G. Teschl, Jacobi Operators and Completely, Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, Vol 72, Amer. Math. Soc., 2000.  Google Scholar

[1]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4531-4543. doi: 10.3934/dcds.2021047
[2]

Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541
[3]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013
[4]

Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365
[5]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2559-2599. doi: 10.3934/dcds.2020375
[6]

Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems & Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001
[7]

Leyter Potenciano-Machado, Alberto Ruiz. Stability estimates for a magnetic Schrödinger operator with partial data. Inverse Problems & Imaging, 2018, 12 (6) : 1309-1342. doi: 10.3934/ipi.2018055
[8]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[9]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[10]

Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221
[11]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[12]

Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207
[13]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[14]

Ping-Liang Huang, Youde Wang. Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1775-1795. doi: 10.3934/dcdss.2016074
[15]

Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013
[16]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436
[17]

A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
[18]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[19]

Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763
[20]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

 Impact Factor: 

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences