Research Article
BibTex RIS Cite
Year 2020, Volume: 49 Issue: 4, 1234 - 1244, 06.08.2020
https://doi.org/10.15672/hujms.542995

Abstract

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.

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

Year 2020, Volume: 49 Issue: 4, 1234 - 1244, 06.08.2020
https://doi.org/10.15672/hujms.542995

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. 

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.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Elgiz Baıramov 0000-0003-2075-5016

Seyda Solmaz This is me 0000-0001-7572-2655

Serifenur Cebesoy 0000-0003-3571-6386

Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Baıramov, E., Solmaz, S., & Cebesoy, S. (2020). P, T, and PT−symmetries of impulsive Dirac systems. Hacettepe Journal of Mathematics and Statistics, 49(4), 1234-1244. https://doi.org/10.15672/hujms.542995
AMA Baıramov E, Solmaz S, Cebesoy S. P, T, and PT−symmetries of impulsive Dirac systems. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1234-1244. doi:10.15672/hujms.542995
Chicago Baıramov, Elgiz, Seyda Solmaz, and Serifenur Cebesoy. “P, T, and PT−symmetries of Impulsive Dirac Systems”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1234-44. https://doi.org/10.15672/hujms.542995.
EndNote Baıramov E, Solmaz S, Cebesoy S (August 1, 2020) P, T, and PT−symmetries of impulsive Dirac systems. Hacettepe Journal of Mathematics and Statistics 49 4 1234–1244.
IEEE E. Baıramov, S. Solmaz, and S. Cebesoy, “P, T, and PT−symmetries of impulsive Dirac systems”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1234–1244, 2020, doi: 10.15672/hujms.542995.
ISNAD Baıramov, Elgiz et al. “P, T, and PT−symmetries of Impulsive Dirac Systems”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1234-1244. https://doi.org/10.15672/hujms.542995.
JAMA Baıramov E, Solmaz S, Cebesoy S. P, T, and PT−symmetries of impulsive Dirac systems. Hacettepe Journal of Mathematics and Statistics. 2020;49:1234–1244.
MLA Baıramov, Elgiz et al. “P, T, and PT−symmetries of Impulsive Dirac Systems”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1234-4, doi:10.15672/hujms.542995.
Vancouver Baıramov E, Solmaz S, Cebesoy S. P, T, and PT−symmetries of impulsive Dirac systems. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1234-4.