Year 2019, Volume 7, Issue 1, Pages 128 - 135 2019-04-15

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

Bilender P. Allahverdiev [1] , Hüseyin Tuna [2]

19 78

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.



Time scales, Dirac operator, singular point, parseval equality
  • [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] 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] B. P. Allahverdiev and H. Tuna, Spectral expansion for the singular Dirac system with impulsive conditions, Turk J Math, 42, (2018), 2527 – 2545.
  • [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] 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] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2001.
  • [7] M. Bohner and A. Peterson, (Eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, 2003.
  • [8] T. Gulsen and E. Yilmaz, Spectral theory of Dirac system on time scales, Applicable Analysis, 96(16), (2017), 2684–2694.
  • [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] G. Sh. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales. Int. J. Difference Equ. 2 (2007), no. 1, 93–104.
  • [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] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18; (1990); 18􀀀56:
  • [13] G. Hovhannisyan, On Dirac equation on a time scale, Journal of Math. Physics, 52, no.10, 102701, 2011.
  • [14] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis. Translated by R.A. Silverman, Dover Publications, New York, 1970.
  • [15] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996.
  • [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] B. P. Rynne, L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328; (2007); 1217􀀀1236.
  • [18] B. Thaller, The Dirac Equation, Springer, 1992.
  • [19] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I. Second Edition Clarendon Press, Oxford, 1962.
  • [20] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, 1258, Springer, Berlin 1987.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: Bilender P. Allahverdiev
Institution: SÜLEYMAN DEMİREL ÜNİVERSİTESİ, FEN-EDEBİYAT FAKÜLTESİ, MATEMATİK BÖLÜMÜ
Country: Turkey


Author: Hüseyin Tuna (Primary Author)
Institution: MEHMET AKİF ERSOY ÜNİVERSİTESİ, FEN-EDEBİYAT FAKÜLTESİ, MATEMATİK BÖLÜMÜ
Country: Turkey


Dates

Publication Date: April 15, 2019

Bibtex @research article { konuralpjournalmath450635, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2019}, volume = {7}, pages = {128 - 135}, doi = {}, title = {Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales}, key = {cite}, author = {Allahverdiev, Bilender P. and Tuna, Hüseyin} }
APA Allahverdiev, B , Tuna, H . (2019). Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp Journal of Mathematics (KJM), 7 (1), 128-135. Retrieved from http://dergipark.org.tr/konuralpjournalmath/issue/31492/450635
MLA Allahverdiev, B , Tuna, H . "Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales". Konuralp Journal of Mathematics (KJM) 7 (2019): 128-135 <http://dergipark.org.tr/konuralpjournalmath/issue/31492/450635>
Chicago Allahverdiev, B , Tuna, H . "Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales". Konuralp Journal of Mathematics (KJM) 7 (2019): 128-135
RIS TY - JOUR T1 - Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales AU - Bilender P. Allahverdiev , Hüseyin Tuna Y1 - 2019 PY - 2019 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 128 EP - 135 VL - 7 IS - 1 SN - -2147-625X M3 - UR - Y2 - 2018 ER -
EndNote %0 Konuralp Journal of Mathematics (KJM) Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales %A Bilender P. Allahverdiev , Hüseyin Tuna %T Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales %D 2019 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 7 %N 1 %R %U
ISNAD Allahverdiev, Bilender P. , Tuna, Hüseyin . "Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales". Konuralp Journal of Mathematics (KJM) 7 / 1 (April 2019): 128-135.
AMA Allahverdiev B , Tuna H . Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp J. Math.. 2019; 7(1): 128-135.
Vancouver Allahverdiev B , Tuna H . Eigenfunction Expansion in the Singular Case for Dirac Systems on Time Scales. Konuralp Journal of Mathematics (KJM). 2019; 7(1): 135-128.