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Year 2023, Volume: 11 Issue: 2, 218 - 228, 31.10.2023

Abstract

References

  • [1] M. Adıvar and E. Bairamov, Spectral properties of non-selfadjoint difference operators, Journal of Mathematical Analysis and Applications, Vol:261, No.2 (2001), 461-478.
  • [2] M. Adıvar and M. Bohner, Spectral analysis of q-difference equations with spectral singularities, Mathematical and Computer Modeling, Vol:43, No.7-8 (2006), 695-703.
  • [3] H. Adil and E. Bairamov, An eigenvalue problem for quadratic pencils of q-difference equations and its applications, Applied Mathematics Letters, Vol:22, No.4 (2009), 521-527.
  • [4] Y. Aygar, Quadratic eigenparameter-dependent quantum difference equations, Turkish Journal of Mathematics, Vol:40, No.2 (2016), 445-452.
  • [5] Y. Aygar, Investigation of spectral analysis of matrix quantum difference equations with spectral singularities, Hacettepe Journal of Mathematics and Statistics, Vol:45, No.4 (2016), 999-1005.
  • [6] Y. Aygar, A Research on spectral analysis of a matrix quantum difference equations with spectral singularities, Quaestiones Mathematicae, Vol:40, No.2 (2017), 245-249.
  • [7] Y. Aygar and E. Bairamov, Scattering theory of impulsive Sturm-Liouville equation in quantum calculus, Bulletin of the Malaysian Mathematical Sciences Society, Vol:42, No.6 (2019), 3247-3259.
  • [8] Y. Aygar and M. Bohner, On the spectrum of eigenparameter-dependent quantum difference equations, Applied Mathematics and Information Sciences, Vo:9, No.4 (2015), 1-5.
  • [9] Y. Aygar and M. Bohner, A polynomial-type Jost solution and spectral properties of a self-adjoint quantum-difference operator, Complex Analysis and Operator Theory, Vol:10, No.6 (2016), 1171-1180.
  • [10] Y. Aygar and M. Bohner, Spectral analysis of a matrix-valued quantum-difference operator, Dynamic Systems and Applications, Vol:25, No.1-2 (2016), 29-37.
  • [11] Y. Aygar and G. G.O¨ zbey, Scattering analysis of a quantum impulsive boundary value problem with spectral parameter, Hacettepe Journal ofMathematics and Statistics, Vol:51, No.1 (2022), 142-155.
  • [12] Bainov D. and Simeonov P. S., Impulsive Differential Equations, Periodic Solutions and Applications, Harlow: Longman Scientific and Technical, 1993.
  • [13] E. M. Bairamov, A condition for finiteness of the discrete spectrum of a second-order nonselfadjoint difference operator on the semi-axis, Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk, Vol: 4, (1984), 13-18.
  • [14] E. M. Bairamov, Structure of the spectrum of a system of a first-order nonselfadjoint difference operator on the half-line, Spectral Theory of Operators and Its Applications, Vol:9, (1989), 55-58.
  • [15] E. Bairamov, S. Cebesoy and I. Erdal, Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operator, Journal of Applied Analysis and Computation, Vol:9, No.4 (2019), 1454-1469.
  • [16] E. Bairamov and A. O. Celebi, Spectrum and spectral expansion for the nonselfadjoint discrete Dirac operators, Quarterly Journal of Mathematics, Vol:50, No.200 (1999),371-384.
  • [17] E. Bairamov and C. Coskun, Jost solutions and the spectrum of the system of difference equations, Applied Mathematics Letters, Vol:17, No.9 (2004), 1039-1045.
  • [18] E. Bairamov, O¨ . C¸ akar and A. M. Krall, Nonselfadjoint difference operators and Jacobi matrices with spectral singularities, Mathematische Nachrichten, Vol:229, No.1 (2001), 5-14.
  • [19] E. Bairamov and T. Koprubasi, Eigenparameter dependent discrete Dirac equations with spectral singularities, Applied Mathematics and Computation, Vol:215, No.12 (2010), 4216-4220.
  • [20] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001.
  • [21] Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2002.
  • [22] E. P. Dolzhenko, Boundary-value uniqueness theorems for analytic functions, Mathematical Notes of the Academy of Sciences of the USSR, Vol: 25, No.6 (1979), 437-442.
  • [23] Ernst, T., The history of q-Calculus and a New Method, Department of Mathematics, Sweden, 2000.
  • [24] Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Jerusalem, Israel Program for Scientific Translations, 2000.
  • [25] T. Gulsen, I. Jadlovska and E. Yılmaz, On the number of eigenvalues for parameter-dependent diffusion problem on time scales, Mathemaical Methods in the Applied Sciences, Vol:44, No.1 (2021), 985-992.
  • [26] Hilger, S., Ein Masskettenkalk¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universit¨at W¨urzburg, 1988.
  • [27] Kac, V., Cheung, P., Quantum Calculus, New York, Springer, 2002.
  • [28] A. M. Krall, E. Bairamov and O. Cakar, Spectral analysis of nonselfadjoint discrete Schr¨odinger operators with spectral singularities, Mathematische Nachrichten, Vol:231, No.1 (2001), 89-104.
  • [29] Lakshmikantham, V. and Bainov, D., Simeonov, P. S., Theory of Impulsive Differential Equations, Teaneck, NJ, World Scientific, 1989.
  • [30] S. Lewanowicz, Construction of recurrences for the coefficients of expansions in q-classical orthogonal polynomials, Journal of Computational and Applied Mathematics, Vol:153, (2003), 295-309.
  • [31] Lusternik, L. A. and Sobolev, V. J., Elements of Functional Analysis, New York, Halsted Press, 1968.
  • [32] X. Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Applied Mathematics Letters, Vol:25, No.2 (2012), 133-137.
  • [33] F. G. Maksudov, B. P. Allahverdiev and E. M. Bairamov, On the spectral theory of a nonselfadjoint operator generated by an infinite Jacobi matrix, Doklady Akademii Nauk, Vol:316, No.2 (1991), 292-296.
  • [34] Naimark, M. A., Investigation of the Spectrum and the Expansion in Eigenfunction of a Nonselfadjoint Operator of Second Order on a Semi-Axis, American Mathematical Society Translations, 1960.
  • [35] J. Nieto and D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Analysis, Real World Applications, Vol: 10, No.2 (2009), 680-690.
  • [36] Samoilenko, A. M. and Perestyuk, N. A., Impulsive Differential Equations, Singapore, World Scientific Publishing Corporation, 1995.
  • [37] W. J. Trjitzinsky, Analytic theory of linear q-difference equations, Acta Mathematica, Vol:61, (1933), 1-38.
  • [38] E. Yılmaz, H. Koyunbakan and U. Ic¸, Some spectral properties of diffusion equation on time scales, Contemporary Analysis and Applied Mathematics, Vol:3, No.2 (2015), 238-246.
  • [39] X. M. Zheng and Z. X. Chen, Some properties of meromorphic solutions of q-difference equations, Journal of Mathematical Analysis and Applications, Vol:361, No.2 (2010), 472-480.

Quantum Difference Problem with Point Interaction

Year 2023, Volume: 11 Issue: 2, 218 - 228, 31.10.2023

Abstract

The main aim of this study is to examine the spectral analysis of q-difference equation with point interaction. We first find Jost solution and Jost function of this problem. Next, we establish the resolvent operator, continuous spectrum and discrete spectrum of the problem. At last, we demonstrate that the quantum boundary value problem with point interaction has finite number of eigenvalues and spectral singularities with finite multiplicities under certain conditions.

References

  • [1] M. Adıvar and E. Bairamov, Spectral properties of non-selfadjoint difference operators, Journal of Mathematical Analysis and Applications, Vol:261, No.2 (2001), 461-478.
  • [2] M. Adıvar and M. Bohner, Spectral analysis of q-difference equations with spectral singularities, Mathematical and Computer Modeling, Vol:43, No.7-8 (2006), 695-703.
  • [3] H. Adil and E. Bairamov, An eigenvalue problem for quadratic pencils of q-difference equations and its applications, Applied Mathematics Letters, Vol:22, No.4 (2009), 521-527.
  • [4] Y. Aygar, Quadratic eigenparameter-dependent quantum difference equations, Turkish Journal of Mathematics, Vol:40, No.2 (2016), 445-452.
  • [5] Y. Aygar, Investigation of spectral analysis of matrix quantum difference equations with spectral singularities, Hacettepe Journal of Mathematics and Statistics, Vol:45, No.4 (2016), 999-1005.
  • [6] Y. Aygar, A Research on spectral analysis of a matrix quantum difference equations with spectral singularities, Quaestiones Mathematicae, Vol:40, No.2 (2017), 245-249.
  • [7] Y. Aygar and E. Bairamov, Scattering theory of impulsive Sturm-Liouville equation in quantum calculus, Bulletin of the Malaysian Mathematical Sciences Society, Vol:42, No.6 (2019), 3247-3259.
  • [8] Y. Aygar and M. Bohner, On the spectrum of eigenparameter-dependent quantum difference equations, Applied Mathematics and Information Sciences, Vo:9, No.4 (2015), 1-5.
  • [9] Y. Aygar and M. Bohner, A polynomial-type Jost solution and spectral properties of a self-adjoint quantum-difference operator, Complex Analysis and Operator Theory, Vol:10, No.6 (2016), 1171-1180.
  • [10] Y. Aygar and M. Bohner, Spectral analysis of a matrix-valued quantum-difference operator, Dynamic Systems and Applications, Vol:25, No.1-2 (2016), 29-37.
  • [11] Y. Aygar and G. G.O¨ zbey, Scattering analysis of a quantum impulsive boundary value problem with spectral parameter, Hacettepe Journal ofMathematics and Statistics, Vol:51, No.1 (2022), 142-155.
  • [12] Bainov D. and Simeonov P. S., Impulsive Differential Equations, Periodic Solutions and Applications, Harlow: Longman Scientific and Technical, 1993.
  • [13] E. M. Bairamov, A condition for finiteness of the discrete spectrum of a second-order nonselfadjoint difference operator on the semi-axis, Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk, Vol: 4, (1984), 13-18.
  • [14] E. M. Bairamov, Structure of the spectrum of a system of a first-order nonselfadjoint difference operator on the half-line, Spectral Theory of Operators and Its Applications, Vol:9, (1989), 55-58.
  • [15] E. Bairamov, S. Cebesoy and I. Erdal, Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operator, Journal of Applied Analysis and Computation, Vol:9, No.4 (2019), 1454-1469.
  • [16] E. Bairamov and A. O. Celebi, Spectrum and spectral expansion for the nonselfadjoint discrete Dirac operators, Quarterly Journal of Mathematics, Vol:50, No.200 (1999),371-384.
  • [17] E. Bairamov and C. Coskun, Jost solutions and the spectrum of the system of difference equations, Applied Mathematics Letters, Vol:17, No.9 (2004), 1039-1045.
  • [18] E. Bairamov, O¨ . C¸ akar and A. M. Krall, Nonselfadjoint difference operators and Jacobi matrices with spectral singularities, Mathematische Nachrichten, Vol:229, No.1 (2001), 5-14.
  • [19] E. Bairamov and T. Koprubasi, Eigenparameter dependent discrete Dirac equations with spectral singularities, Applied Mathematics and Computation, Vol:215, No.12 (2010), 4216-4220.
  • [20] Bohner, M. and Peterson, A., Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001.
  • [21] Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2002.
  • [22] E. P. Dolzhenko, Boundary-value uniqueness theorems for analytic functions, Mathematical Notes of the Academy of Sciences of the USSR, Vol: 25, No.6 (1979), 437-442.
  • [23] Ernst, T., The history of q-Calculus and a New Method, Department of Mathematics, Sweden, 2000.
  • [24] Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Jerusalem, Israel Program for Scientific Translations, 2000.
  • [25] T. Gulsen, I. Jadlovska and E. Yılmaz, On the number of eigenvalues for parameter-dependent diffusion problem on time scales, Mathemaical Methods in the Applied Sciences, Vol:44, No.1 (2021), 985-992.
  • [26] Hilger, S., Ein Masskettenkalk¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universit¨at W¨urzburg, 1988.
  • [27] Kac, V., Cheung, P., Quantum Calculus, New York, Springer, 2002.
  • [28] A. M. Krall, E. Bairamov and O. Cakar, Spectral analysis of nonselfadjoint discrete Schr¨odinger operators with spectral singularities, Mathematische Nachrichten, Vol:231, No.1 (2001), 89-104.
  • [29] Lakshmikantham, V. and Bainov, D., Simeonov, P. S., Theory of Impulsive Differential Equations, Teaneck, NJ, World Scientific, 1989.
  • [30] S. Lewanowicz, Construction of recurrences for the coefficients of expansions in q-classical orthogonal polynomials, Journal of Computational and Applied Mathematics, Vol:153, (2003), 295-309.
  • [31] Lusternik, L. A. and Sobolev, V. J., Elements of Functional Analysis, New York, Halsted Press, 1968.
  • [32] X. Li, Further analysis on uniform stability of impulsive infinite delay differential equations, Applied Mathematics Letters, Vol:25, No.2 (2012), 133-137.
  • [33] F. G. Maksudov, B. P. Allahverdiev and E. M. Bairamov, On the spectral theory of a nonselfadjoint operator generated by an infinite Jacobi matrix, Doklady Akademii Nauk, Vol:316, No.2 (1991), 292-296.
  • [34] Naimark, M. A., Investigation of the Spectrum and the Expansion in Eigenfunction of a Nonselfadjoint Operator of Second Order on a Semi-Axis, American Mathematical Society Translations, 1960.
  • [35] J. Nieto and D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Analysis, Real World Applications, Vol: 10, No.2 (2009), 680-690.
  • [36] Samoilenko, A. M. and Perestyuk, N. A., Impulsive Differential Equations, Singapore, World Scientific Publishing Corporation, 1995.
  • [37] W. J. Trjitzinsky, Analytic theory of linear q-difference equations, Acta Mathematica, Vol:61, (1933), 1-38.
  • [38] E. Yılmaz, H. Koyunbakan and U. Ic¸, Some spectral properties of diffusion equation on time scales, Contemporary Analysis and Applied Mathematics, Vol:3, No.2 (2015), 238-246.
  • [39] X. M. Zheng and Z. X. Chen, Some properties of meromorphic solutions of q-difference equations, Journal of Mathematical Analysis and Applications, Vol:361, No.2 (2010), 472-480.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Yelda Aygarküçükevcilioğlu

Güher Gülçehre Özbey 0000-0002-1326-4545

Güler Başak Öznur 0000-0003-4130-5348

Publication Date October 31, 2023
Submission Date May 24, 2023
Acceptance Date October 4, 2023
Published in Issue Year 2023 Volume: 11 Issue: 2

Cite

APA Aygarküçükevcilioğlu, Y., Özbey, G. G., & Öznur, G. B. (2023). Quantum Difference Problem with Point Interaction. Konuralp Journal of Mathematics, 11(2), 218-228.
AMA Aygarküçükevcilioğlu Y, Özbey GG, Öznur GB. Quantum Difference Problem with Point Interaction. Konuralp J. Math. October 2023;11(2):218-228.
Chicago Aygarküçükevcilioğlu, Yelda, Güher Gülçehre Özbey, and Güler Başak Öznur. “Quantum Difference Problem With Point Interaction”. Konuralp Journal of Mathematics 11, no. 2 (October 2023): 218-28.
EndNote Aygarküçükevcilioğlu Y, Özbey GG, Öznur GB (October 1, 2023) Quantum Difference Problem with Point Interaction. Konuralp Journal of Mathematics 11 2 218–228.
IEEE Y. Aygarküçükevcilioğlu, G. G. Özbey, and G. B. Öznur, “Quantum Difference Problem with Point Interaction”, Konuralp J. Math., vol. 11, no. 2, pp. 218–228, 2023.
ISNAD Aygarküçükevcilioğlu, Yelda et al. “Quantum Difference Problem With Point Interaction”. Konuralp Journal of Mathematics 11/2 (October 2023), 218-228.
JAMA Aygarküçükevcilioğlu Y, Özbey GG, Öznur GB. Quantum Difference Problem with Point Interaction. Konuralp J. Math. 2023;11:218–228.
MLA Aygarküçükevcilioğlu, Yelda et al. “Quantum Difference Problem With Point Interaction”. Konuralp Journal of Mathematics, vol. 11, no. 2, 2023, pp. 218-2.
Vancouver Aygarküçükevcilioğlu Y, Özbey GG, Öznur GB. Quantum Difference Problem with Point Interaction. Konuralp J. Math. 2023;11(2):218-2.
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