In this work, we study the one-dimensional Dirac system on a whole line with impulsive conditions. We construct a spectral function of this system. Using the spectral function, we establish a Parseval equality and spectral expansion formula for such a system. | Turk J Math (2018) 42: 2527 – 2545 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Spectral expansion for the singular Dirac system with impulsive conditions Bilender PAŞAOĞLU ALLAHVERDİEV1 ,, Hüseyin TUNA2,∗, Department of Mathematics, Faculty of Science, Süleyman Demirel University, Isparta, Turkey 2 Department of Mathematics, Faculty of Science, Mehmet Akif Ersoy University, Burdur, Turkey 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this work, we study the one-dimensional Dirac system on a whole line with impulsive conditions. We construct a spectral function of this system. Using the spectral function, we establish a Parseval equality and spectral expansion formula for such a system. Key words: Dirac operator, impulsive conditions, singular point, spectral function, Parseval equality, spectral expansion 1. Introduction Many problems of engineering interest are governed by partial differential equations. When we seek a solution of a partial differential equation by separation of variables, it leads to the problem of expanding an arbitrary function as a series of eigenfunctions. The method relies on the completeness of the eigenfunctions corresponding to one of the variables. Thus, spectral expansion theorems are important in mathematics. Using several methods, the eigenfunction expansion is obtained, including the methods of integral equations, contour integration, and finite difference (see [19], [32]). The Dirac operators play an important role in the theory of relativistic quantum mechanics because fundamental physics of relativistic quantum mechanics was formulated by the Dirac operators. For example, they predict the existence of a positron and elucidate the origin of spin 1/2 of an electron. We refer the reader to [30]. Discontinuous (or impulsive) boundary value problems have been extensively investigated in recent .
