P, T, and PT−symmetries of impulsive Dirac systems

Year 2020, Volume: 49 Issue: 4, 1234 - 1244, 06.08.2020

Elgiz Baıramov , Seyda Solmaz Serifenur Cebesoy

https://doi.org/10.15672/hujms.542995

Cited By: 1

https://izlik.org/JA54JF87XD

Abstract

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 P, T, and PT−symmetry. 

Keywords

Impulsive operators , bound states , spectral singularities , PT- symmetry

References

• [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.