$mathcal{P}$, $mathcal{T}$, and $mathcal{P}$$mathcal{T}$−symmetries of impulsive Dirac

This article is concerned with locations of bound states and spectral singularities of an impulsive Dirac system. By using a transfer matrix, we obtain some spectral properties of this impulsive system. We also examine some special cases, where the impulsive condition at the origin has $mathcal{P}$, $mathcal{T}$, and $mathcal{P}$$mathcal{T}$−symmetry.

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[1] Y. Aygar and M. Olgun, Investigation of the spectrum and the Jost solutions of discrete Dirac system on the whole axis, J. Inequal. Appl. 2014, Art. No. 73, 2014.
[2] E. Bairamov, O. Cakar, and A.M. Krall, An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities, J. Differential Equations, 151 (2), 268–289, 1999.
[3] E. Bairamov and A.O. Celebi, Spectrum and spectral expansion for the nonselfadjoint discrete Dirac operators, Quart. J. Math. Oxford Ser. 50 (200), 371–384, 1999.
[4] E. Bairamov and O. Karaman, Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition, Acta Math. Hungar. 97 (1-2), 121–131, 2002.
[5] C.M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (24), 5243–5246, 1998.
[6] G.Sh. Guseinov, On the concept of spectral singularities, Pramana J. Phys. 73 (3), 587–603, 2009.
[7] A.M. Krall, E. Bairamov, and O. Cakar, Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition, J. Differential Equations, 151 (2), 252–267, 1999.
[8] V. Lakshmikantham, D.D. Bainov, and P.S. Simeonov, Theory of Impulsive Differential Equations 6, in: Series in Modern Applied Mathematics, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
[9] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators 59, in: Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.
[10] V.E. Lyance, On a differential operator with spectral singularities, AMS Transl. I ,II 60 (2), 185–225, 227–283, 1967.
[11] A. Mostafazadeh, Spectral singularities of a general point interaction, J. Phys. A 44 (37), 375302, 2011.
[12] A. Mostafazadeh and H.M. Dehnavi, Spectral singularities, biorthonormal systems and a two-parameter family of complex point interactions, J. Phys. A 42 (12), 125303, 2009.
[13] O.Sh. Mukhtarov and K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Math. Sci. Ser. B Engl. Ed. 35 (3), 639–649, 2015.
[14] O.Sh. Mukhtarov, H. Olgar, and K. Aydemir, Resolvent operator and spectrum of new type boundary value problems, Filomat 29 (7), 1671–1680, 2015.
[15] B. Nagy, Operators with spectral singularities, J. Operator Theory 15 (2), 307–325, 1986.
[16] M.A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, Amer. Math. Soc. Transl. 16 (2), 103–193, 1960.
[17] H. Olgar, O.Sh. Mukhtarov, and K. Aydemir, Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem, Filomat 32 (3), 911–920, 2018.
[18] B.S. Pavlov, On the spectral theory of non-selfadjoint differential operators, Dokl. Akad. Nauk SSSR 146, 1267–1270, 1962.
[19] J. Schwartz, Some non-selfadjoint operators, Comm. Pure Appl. Math. 13, 609–639, 1960.
[20] E. Ugurlu, On the perturbation determinants of a singular dissipative boundary value problem with finite transmission conditions, J. Math. Anal. Appl. 409 (1), 567–575, 2014.
[21] E. Ugurlu and E. Bairamov. Krein’s theorem for the dissipative operators with finite impulsive effects, Numer. Funct. Anal. Optim. 36 (2), 256–270, 2015.
[22] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1996.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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