Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales

Abstract

In this work, we prove the existence of a spectral function for one dimensional singular Dirac operator on time scales. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.

Keywords

• Time scales,Dirac operator,singular point,parseval equality

References

1. [1] R. P. Agarwal, M. Bohner and D. O’Regan, Time scale boundary value problems on infinite intervals, J. Comput. Appl. Math., 141 (2002), 27-34.
2. [2] B. P. Allahverdiev and H. Tuna, An expansion theorem for q-Sturm-Liouville operators on the whole line, Turk J Math, 42, (2018), 1060-1071.
3. [3] B. P. Allahverdiev and H. Tuna, Spectral expansion for the singular Dirac system with impulsive conditions, Turk J Math, 42, (2018), 2527 – 2545.
4. [4] D. R. Anderson, G. Sh. Guseinov and J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194 (2) (2006) ; 309-342.
5. [5] F. Atici Merdivenci and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141 (1-2) (2002); 75-99.
6. [6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2001.
7. [7] M. Bohner and A. Peterson, (Eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
8. [8] T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Applicable Analysis, 96(16), (2017), 2684–2694.

9. [9] G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29 (4); (2005) ; 365􀀀380.
10. [10] G. Sh. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equ. 2 (2007), no. 1, 93–104.
11. [11] G. Sh. Guseinov, An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales. Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 147–160.
12. [12] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18; (1990); 18􀀀56:
13. [13] G. Hovhannisyan, On Dirac equation on a time scale, Journal of Math. Physics, 52, no.10, 102701, 2011.
14. [14] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis. Translated by R.A. Silverman, Dover Publications, New York, 1970.
15. [15] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
16. [16] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian).
17. [17] B. P. Rynne, L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328; (2007); 1217􀀀1236.
18. [18] B. Thaller, The Dirac Equation, Springer, 1992.
19. [19] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition Clarendon Press, Oxford, 1962.
20. [20] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin 1987.