Spectral analysis of Hahn-Dirac system





Hahn-Dirac system, Self-adjoint operator, Eigenvalues and eigenfunctions, Green’s matrix, Eigenfunction expansion


In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Greens function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2w,q ((w0. a): E).

Author Biographies

Bilender Allahverdiev, Süleyman Demirel University.

Dept. of Mathematics,

Hüseyin Tuna, Mehmet Akif Ersoy University.

Dept. of Mathematics.


B. P. Allahverdiev and H. Tuna, “One-dimensional q−Dirac equation”, Mathematical methods in the applied sciences, vol. 40, no. 18, pp. 7287-7306, 2017.

B. P. Allahverdiev and H. Tuna, “The spectral expansion for the Hahn—Dirac system on the whole line”, Turkish journal of mathematics, vol. 43, pp. 1668-1687, 2019.

B. P. Allahverdiev and H. Tuna, “The Parseval equality and expansion formula for Singular Hahn-Dirac system,” in Emerging applications of differential equations and game theory, S. Z. Alparslan Gök and D. Aruğaslan Çinçin, Eds. Hershey, PA: IGI Global, 2019, pp. 209–235.

K. Aldwoah, "Generalized time scales and associated difference equations", Ph.D. Thesis, Cairo University, 2009.

R. Álvarez-Nodarse, “On characterizations of classical polynomials”, Journal of computational and applied mathematics, vol. 196, no. 1, pp. 320-337, 2006.

M. H. Annaby, A. E. Hamza, and K. A. Aldwoah, “Hahn difference operator and associated Jackson-Nörlund integrals”, Journal of optimization theory and applications, vol. 154, pp. 133-153, 2012.

M. H. Annaby, A. E. Hamza, and S. D. Makharesh, “A Sturm-Liouville theory for Hahn difference operator”, in Frontiers of Orthogonal Polynomials and q−Series, X. Li and Z. Nashed, Eds. Singapore: World Scientific, pp. 35-84, 2018.

A. Dobrogowska and A. Odzijewicz, “Second order q−difference equations solvable by factorization method”, Journal of computational and applied mathematics, vol. 193, no. 1, pp. 319-346, 2006.

W. Hahn, ¨Uber orthogonalpolynome, die q−differenzengleichungen genügen”, Mathematische nachrichten, vol. 2, no. 1-2, pp. 4-34, 1949.

W. Hahn, “Ein beitrag zur theorie der orthogonalpolynome”, Monatshefte für mathematik, vol. 95, no. 1, pp. 19-24, 1983.

A. E. Hamza and S. A. Ahmed, “Theory of linear Hahn difference equations”, Journal of advances in mathematics, vol. 4, no. 2, pp. 440-460, 2013.

A. E. Hamza and S. A. Ahmed, “Existence and uniqueness of solutions of Hahn difference equations”, Advances in difference equations, vol. 2013, Art. ID. 316, 2013.

A. E. Hamza and S. D. Makharesh, “Leibniz’s rule and Fubinis theorem associated with Hahn difference operator”, Journal of advanced mathematical, vol. 12, no. 6, pp. 6335-6345, 2016.

F. Hira, “Dirac system associated with Hahn difference operator”, Bulletin of the Malaysian Mathematical Sciences Society, vol. 43, pp. 3481-3497, 2020.

F. H. Jackson, “q−Difference equations”, American journal of mathematics, vol. 32, no. 4, pp. 305-314, 1910.

A. N. Kolmogorov and S. V. Fomin, Introductory real analysis. New York, NY: Dover, 1970.

K. H. Kwon, D. W. Lee, S. B. Park, and B. H. Yoo, “Hahn class orthogonal polynomials”, Kyungpook mathematical journal, vol. 38, pp. 259-281, 1998.

P. Lesky, Eine charakterisierung der Klassischen kontinuierlichen-, diskreten- und Q-orthogonalpolynome. Aachen: Shaker, 2005.

B. M. Levitan and I. S: Sargsjan, Sturm-Liouville and Dirac operators. Dordrecht: Kluwer, 1991.

M. A. Naĭmark, Linear differential operators, 2 vols. New York, NY: F. Ungar, 1969.

J. Petronilho, “Generic formulas for the values at the singular points of some special monic classical Hq,ω−orthogonal polynomials”, Journal of computational and applied mathematics, vol. 205, no. 1, pp. 314-324, 2007.

T. Sitthiwirattham, “On a nonlocal boundary value problem for nonlinear second-order Hahn difference equation with two different q, ω−derivatives”, Advances in difference equations, vol. 116, no. 1, Art. ID. 116, 2016.

B. Thaller, The Dirac equation. Berlin: Springer, 1992.

J. Weidmann, Spectral theory of ordinary differential operators. Berlin: Springer, 1987.



How to Cite

B. Allahverdiev and H. Tuna, “Spectral analysis of Hahn-Dirac system”, Proyecciones (Antofagasta, On line), vol. 40, no. 6, pp. 1547-1567, Apr. 2021.




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