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Singular discrete dirac equations

  • Bilender P. Allahverdiev and Hüseyin Tuna EMAIL logo
Published/Copyright: August 9, 2025
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Abstract

This paper investigates singular discrete Dirac equations and establishes famous Weyls classification for these equations. First, the limit-circle and limit-point classifications are obtained. Then, the existence of boundary conditions is demonstrated. Finally, it is shown that the limit-circle case does not occur for these equations.

  1. (Communicated by Irena Jadlovská)

References

[1] Ahlbrandt, C. D.—Peterson, A. C.: Discrete Hamiltonian Systems, Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences, Kluwer Academic Publishers, Dordrecht, 1996.10.1007/978-1-4757-2467-7Search in Google Scholar

[2] Amirov, R. K.: On a system of Dirac differential equations with discontinuity conditions inside an interval, Ukrain. Math. J. 57 (2005), 712–727.10.1007/s11253-005-0222-7Search in Google Scholar

[3] Aygar, Y.—Olgun, M.—Koprubasi, T.: Principal functions of nonselfadjoint discrete Dirac equations with spectral parameter in boundary conditions, Abstr. Appl. Anal. 2012 (2012), Art. ID 924628.10.1155/2012/924628Search in Google Scholar

[4] Bairamov, E.—Aygar, Y.—Olgun, M.: Jost solution and the spectrum of the discrete Dirac systems, Bound. Value Probl. 2010 (2010), Art. ID 306571.10.1155/2010/306571Search in Google Scholar

[5] Bairamov, E.—Çelebi, A. O.: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators, Quart. J. Math. 50(200) (1999), 371–384.10.1093/qjmath/50.200.371Search in Google Scholar

[6] Berezanskii, J. M.: Expansions in Eigenfunctions of Selfadjoint Operators, Providence, American Mathematical Society, 1968.10.1090/mmono/017Search in Google Scholar

[7] Bohner, M.—Koyunbakan, H.: Inverse problems for Sturm–Liouville difference equations, Filomat 30(5) (2016), 1297–1304.10.2298/FIL1605297BSearch in Google Scholar

[8] Bohner, M.: Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199 (1996), 804–826.10.1006/jmaa.1996.0177Search in Google Scholar

[9] Gulsen, T.—Sian, S. M.—Yilmaz, E.—Koyunbakan, H.: Impulsive diffusion equation on time scales, Int. J. Anal. Appl. 16(1) (2018), 137–148.Search in Google Scholar

[10] Jirari, A.: Second-order Sturm–Liouville difference equations and orthogonal polynomials, Mem. Amer. Math. Soc. 113 (542) (1995), 1–138.10.1090/memo/0542Search in Google Scholar

[11] Levitan, B. M.—Sargsjan, I. S.: Sturm–Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).10.1007/978-94-011-3748-5Search in Google Scholar

[12] Mamedov, K. R.—Akcay, O.: Inverse problem for a class of Dirac operators by the Weyl function, Dyn. Syst. Appl. 26(1) (2017), 183–196.Search in Google Scholar

[13] Mukhtarov, O. S.—Aydemir, K.: The eigenvalue problem with Interaction conditions at one interior singular point, Filomat 31(17) (2017), 5411–5420.10.2298/FIL1717411MSearch in Google Scholar

[14] Mukhtarov, O. S.—Olgar, H.—Aydemir, K.: Resolvent operator and spectrum of new type boundary value problems, Filomat 29(7), (2015), 1671–1680.10.2298/FIL1507671MSearch in Google Scholar

[15] Naimark, M. A.: Linear Differential Operators, 2nd edn., Nauka, Moscow, 1969; English transl. of 1st. edn., 1,2, New York, 1968.Search in Google Scholar

[16] Ozkan, A. S.—Amirov, R. K.: An interior inverse problem for the impulsive Dirac operator, Tamkang J. Math. 42(3) (2011), 259–263.10.5556/j.tkjm.42.2011.824Search in Google Scholar

[17] Shi, Y.: Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems, Linear Algebra Appl. 416 (2006), 452–519.10.1016/j.laa.2005.11.025Search in Google Scholar

[18] Titchmarsh, E. C.: Eigenfunction Expansions Asociated with Second-Order Differential Equations, Part 1, 2nd ed., Clarendon Press, Oxford, 1962.10.1063/1.3058324Search in Google Scholar

[19] Weidmann, J.: Spectral Theory of Ordinary Differential Operators, Lecture Notes in Math., Vol. 1258, Springer, Berlin, 1987.10.1007/BFb0077960Search in Google Scholar

[20] Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Functionen, Math. Ann. 68 (1910), 222–269.10.1007/BF01474161Search in Google Scholar

Received: 2024-12-08
Accepted: 2025-03-14
Published Online: 2025-08-09
Published in Print: 2025-08-26

© 2025 Mathematical Institute Slovak Academy of Sciences

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