$q$-Multiplicative Dirac System

Year 2023, Volume: 11 Issue: 1, 61 - 69, 30.04.2023

Bilender Paşaoğlu , Hüseyin Tuna

Abstract

In this study, the classical Dirac equation was investigated on the basis of $q$-multiplicative calculus. We discuss some spectral properties of the $q$-multiplicative Dirac system, such as formally self-adjointness, and orthogonality of eigenfunctions. Finally, Green's function for this system has been reconstructed.

Keywords

q-multiplicave calculus, Dirac equation., self-adjoint operator

References

• [1] Allahverdiev B. P. and Tuna H., One-dimensional q-Dirac equation, Math. Meth. Appl. Sci.; 402017:7287ñ7306.
• [2] Annaby M. H. and Mansour Z. S., q-Fractional Calculus and Equations, Lecture Notes in Mathematics 2056, Springer, Berlin, 2012.
• [3] Bairamov E., and Solmaz ¸S., Scattering theory of Dirac operator with the impulsive condition on whole axis, Math. Meth. Appl. Sci. 44 (9) (2021), 7732-7746.
• [4] Bashirov A. E., Kurpinar E. M. and Ozyapici A., Multiplicative calculus and its applications, J. Math. Anal.. Appl. 337 (1) (2008),
• [5] Ernst T., The History of q-Calculus and a New Method, U. U. D. M. Report (2000):16, ISSN1101-3591, Department of Mathematics, Uppsala University, 2000.
• [6] Goktas S., Kemaloglu H. and Yilmaz E., Multiplicative conformable fractional Dirac system, Turkish J. Math. 46 (2022), 973-990.
• [7] Goktas S., A new type of SturmñLiouville equation in the nonNewtonian calculus, J. Funct. Spaces 5203939 (2021), 1-8.
• [8] Grossman M., An introduction to Non-Newtonian calculus, Intern. J. Math. Edu. Sci. Techn. 10 (4) (1979), 525-528.
• [9] Grossman M. and Katz R., Non-Newtonian calculus, Pigeon Cove, MA: Lee Press, 1972.
• [10] Gulsen T., Yilmaz E. and Goktas S., Multiplicative Dirac system, Kuwait J. Sci. 2022. doi:10.48129/kjs.13411.
• [11] Kac V. and Cheung P., Quantum Calculus, Springer, 2002.
• [12] Levitan B. M. and Sargsjan I.S., SturmñLiouville and Dirac operators, Math. Appl. (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
• [13] Mamedov, K. R. and Akcay, O., Inverse problem for a class of Dirac operators by the Weyl function, Dyn. Syst. Appl. 26 (1) (2017), 183- 196.
• [14] Mukhtarov O. Sh., Olgar H. and Aydemir K., Resolvent operator and º spectrum of new type boundary value problems, Filomat 29 (7) (2015), 1671-1680.
• [15] Thaller B., The Dirac Equation. Springer-Verlag, Berlin Heidelberg, 1992.
• [16] Weidmann J., Spectral Theory of Ordinary Di§erential Operators, Lecture Notes in Mathematics 1258. Springer: Berlin, 1987.
• [17] Yalcin N. and Celik E., Solution of multiplicative homogeneous linear di§erential equations with constant exponentials, New Trends Math. Sci. 6 (2) (2018), 58-67. 15
• [18] Yener G. and Emiroglu I., A q-analogue of the multiplicative calculus: qmultiplicative calculus, Discr. Contin. Dynam. Syst.-Ser. S 8 (6) (2015), 1435-1450.
• [19] Zettl A., SturmñLiouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society: Providence, RI, 2005.