On the spectral expansion formula for a class of Dirac operators

O. Akcay; Kh. R. Mamedov

doi:10.26637/mjm402/014

Abstract

This paper deals with a problem for the canonical Dirac differential equations system with piecewise continuous coefficient and spectral parameter dependent in boundary conditions. The resolvent operator is constructed. The completeness theorem for eigenvector functions is proved. The spectral expansion formula with respect to eigenvector functions is obtained and Parseval equality is given.

Keywords:

Dirac operator, completeness theorem, expansion formula

Mathematics Subject Classification:

34L10, 34L40

Authors Imprint References Funding Related Articles Downloads

O. Akcay Department of Mathematics, Mersin University, 33343, Mersin, Turkey. https://orcid.org/0000-0001-9691-666X
Kh. R. Mamedov Department of Mathematics, Mersin University, 33343, Mersin, Turkey.

Pages: 297-304
Date Published: 01-04-2016
Vol. 4 No. 02 (2016): Malaya Journal of Matematik (MJM)

H. M. Huseynov and A. R. Latifova, On eigenvalues and eigenfunctions of one class of Dirac operators with discontinuous coefficients, Trans. Natl. Acad. Sci. Azerb. 24(1) (2004) 103-112.

N. B. Kerimov, A boundary value problem for the Dirac system with spectral parameter in the boundary conditions, Differential Equations, 38(2) (2002) 164-174.

V. M. Kurbanov and A. I. Ismailova, Absolute and uniform convergence of expansions in the root vector functions of the Dirac operator, Doklady Mathematics, 86(2) (2012) 663-666. DOI: https://doi.org/10.1134/S1064562412050201

V. M. Kurbanov and A. I. Ismailova, Riesz inequality for systems of root vector functions of the Dirac operator, Differential Equations, 48(3) (2012) 336-342. DOI: https://doi.org/10.1134/S0012266112030044

A. R. Latifova, On the representation of solution with initial conditions for Dirac equations system with discontinuous coefficient, Proceeding of IMM of NAS of Azerbaijan, 16(24) (2002) 64-66.

B. M. Levitan and I. S. Sargsyan, Sturm-Liouville and Dirac Operators, Kluwer Academic Publisher, Dordrecht, Boston, London, 1991. DOI: https://doi.org/10.1007/978-94-011-3748-5

Kh. R. Mamedov and O. Akcay, Inverse problem for a class of Dirac operators with spectral parameter contained in boundary conditions, arXiv:1510.02915v1 [math.SP] (2015).

V. A. Marchenko, Sturm-Liouville Operators and Applications, AMS Chelsea Publishing, Providence, Rhode Island, 2011. DOI: https://doi.org/10.1090/chel/373

G. G. Sahakyan, Oscillation's theorem for one boundary value problem, Taiwanese Journal of Mathematics, 15(5) (2011) 2351-2356. DOI: https://doi.org/10.11650/twjm/1500406439

B. Thaller, The Dirac Equation, Springer, Berlin, 1992. DOI: https://doi.org/10.1007/978-3-662-02753-0

M. M. Tharwat and A. H. Bhrawy, Computation of eigenvalues of discontinuous Dirac system using Hermite interpolation technique, Advances in Difference Equations, 2012:59, doi:10.1186/1687-1847-201259. DOI: https://doi.org/10.1186/1687-1847-2012-59

M. M. Tharwat, A. H. Bhrawy and A. Yildirim, Numerical computation of the eigenvalues of a discontinuous Dirac system using the sinc method with error analysis, International Journal of Computer Mathematics, 89(15) (2012) 2061-2080. DOI: https://doi.org/10.1080/00207160.2012.700112

C.-F. Yang and Z.-Y. Huang, Spectral asymptotics and regularized traces for Dirac operators on a starshaped graph, Applicable Analysis, 91(9) (2012) 1717-1730. DOI: https://doi.org/10.1080/00036811.2011.579563