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Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials

Year 2023, , 29 - 42, 28.03.2023
https://doi.org/10.36753/mathenot.1110497

Abstract

We provide estimates for the periodic and antiperiodic eigenvalues of non-self-adjoint Sturm--Liouville operators with a family of complex-valued trigonometric polynomial potentials. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.

Supporting Institution

Yalova University

Project Number

2019/AP/0010

Thanks

The author is thankful for the Research Fund of Yalova University (Project no. 2019/AP/0010).

References

  • [1] Bagarello, F., Gazeau, J.-P., Szafraniec, F., Znojil, M. (Eds.): Non-selfadjoint operators in quantum physics: Mathematical aspects. JohnWiley & Sons (2015).
  • [2] Bender, C. M.: PT-symmetric potentials having continuous spectra. Journal of Physics A-Mathematical and Theoretical. 53 (37), 375302 (2020).
  • [3] Mostafazadeh, A.: Psevdo-hermitian representation of quantum mechanics. International Journal of Geometric Methods in Modern Physics. 11, 1191-1306 (2010).
  • [4] Veliev, O. A.: On the spectral properties of the Schrodinger operator with a periodic PT-symmetric potential. International Journal of Geometric Methods in Modern Physics. 14, 1750065 (2017).
  • [5] Veliev, O. A.: On the finite-zone periodic PT-symmetric potentials. Moscow Mathematical Journal. 19 (4), 807-816 (2019).
  • [6] Veliev, O. A.: Non-self-adjoint Schrödinger operator with a periodic potential. Springer, Cham (2021).
  • [7] Bender, C. M., Dunne, G. V., Meisinger, P. N.: Complex periodic potentials with real band spectra. Physics Letters A. 252, 272-276 (1999).
  • [8] Brown, B.M., Eastham, M. S. P., Schmidt, K. M.: Periodic differential operators, Operator Theory: Advances and Applications, 230, Birkhuser/Springer: Basel AG, Basel (2013).
  • [9] Levy, M., Keller, B.: Instability intervals of Hill’s equation. Communications on Pure and Applied Mathematics. 16, 469-476 (1963).
  • [10] Magnus,W., Winkler, S.: Hill’s equation. Interscience Publishers, New York (1966).
  • [11] Marchenko, V.: Sturm-Liouville operators and applications. Birkhauser Verlag, Basel (1986).
  • [12] Eastham, M. S. P.: The spectral theory of periodic differential operators. Hafner. New York (1974).
  • [13] Gasymov, M. G.: Spectral analysis of a class of second-order nonself-adjoint differential operators. Fankts. Anal. Prilozhen. 14, 14-19 (1980).
  • [14] Kerimov, N. B.: On a boundary value problem of N. I. Ionkin type. Differential Equations. 49, 1233-1245 (2013).
  • [15] Nur, C.: On the estimates of periodic eigenvalues of Sturm-Liouville operators with trigonometric polynomial potentials. Mathematical Notes. 109 (5), 794-807 (2021).
  • [16] Veliev, O. A.: Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators. Journal of Mathematical Analysis and Applications. 422, 1390-1401 (2015).
  • [17] Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, Boston, Mass, USA (1987).
  • [18] Dernek, N., Veliev, O. A.: On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operators. Israel Journal of Mathematics. 145, 113-123 (2005).
  • [19] Veliev, O. A.: The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential. International Journal of Geometric Methods in Modern Physics. 15, 1850008 (2018).
Year 2023, , 29 - 42, 28.03.2023
https://doi.org/10.36753/mathenot.1110497

Abstract

Project Number

2019/AP/0010

References

  • [1] Bagarello, F., Gazeau, J.-P., Szafraniec, F., Znojil, M. (Eds.): Non-selfadjoint operators in quantum physics: Mathematical aspects. JohnWiley & Sons (2015).
  • [2] Bender, C. M.: PT-symmetric potentials having continuous spectra. Journal of Physics A-Mathematical and Theoretical. 53 (37), 375302 (2020).
  • [3] Mostafazadeh, A.: Psevdo-hermitian representation of quantum mechanics. International Journal of Geometric Methods in Modern Physics. 11, 1191-1306 (2010).
  • [4] Veliev, O. A.: On the spectral properties of the Schrodinger operator with a periodic PT-symmetric potential. International Journal of Geometric Methods in Modern Physics. 14, 1750065 (2017).
  • [5] Veliev, O. A.: On the finite-zone periodic PT-symmetric potentials. Moscow Mathematical Journal. 19 (4), 807-816 (2019).
  • [6] Veliev, O. A.: Non-self-adjoint Schrödinger operator with a periodic potential. Springer, Cham (2021).
  • [7] Bender, C. M., Dunne, G. V., Meisinger, P. N.: Complex periodic potentials with real band spectra. Physics Letters A. 252, 272-276 (1999).
  • [8] Brown, B.M., Eastham, M. S. P., Schmidt, K. M.: Periodic differential operators, Operator Theory: Advances and Applications, 230, Birkhuser/Springer: Basel AG, Basel (2013).
  • [9] Levy, M., Keller, B.: Instability intervals of Hill’s equation. Communications on Pure and Applied Mathematics. 16, 469-476 (1963).
  • [10] Magnus,W., Winkler, S.: Hill’s equation. Interscience Publishers, New York (1966).
  • [11] Marchenko, V.: Sturm-Liouville operators and applications. Birkhauser Verlag, Basel (1986).
  • [12] Eastham, M. S. P.: The spectral theory of periodic differential operators. Hafner. New York (1974).
  • [13] Gasymov, M. G.: Spectral analysis of a class of second-order nonself-adjoint differential operators. Fankts. Anal. Prilozhen. 14, 14-19 (1980).
  • [14] Kerimov, N. B.: On a boundary value problem of N. I. Ionkin type. Differential Equations. 49, 1233-1245 (2013).
  • [15] Nur, C.: On the estimates of periodic eigenvalues of Sturm-Liouville operators with trigonometric polynomial potentials. Mathematical Notes. 109 (5), 794-807 (2021).
  • [16] Veliev, O. A.: Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators. Journal of Mathematical Analysis and Applications. 422, 1390-1401 (2015).
  • [17] Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, Boston, Mass, USA (1987).
  • [18] Dernek, N., Veliev, O. A.: On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operators. Israel Journal of Mathematics. 145, 113-123 (2005).
  • [19] Veliev, O. A.: The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential. International Journal of Geometric Methods in Modern Physics. 15, 1850008 (2018).
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Cemile Nur 0000-0001-7375-3474

Project Number 2019/AP/0010
Publication Date March 28, 2023
Submission Date April 28, 2022
Acceptance Date August 2, 2022
Published in Issue Year 2023

Cite

APA Nur, C. (2023). Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials. Mathematical Sciences and Applications E-Notes, 11(1), 29-42. https://doi.org/10.36753/mathenot.1110497
AMA Nur C. Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials. Math. Sci. Appl. E-Notes. March 2023;11(1):29-42. doi:10.36753/mathenot.1110497
Chicago Nur, Cemile. “Computing Eigenvalues of Sturm--Liouville Operators With a Family of Trigonometric Polynomial Potentials”. Mathematical Sciences and Applications E-Notes 11, no. 1 (March 2023): 29-42. https://doi.org/10.36753/mathenot.1110497.
EndNote Nur C (March 1, 2023) Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials. Mathematical Sciences and Applications E-Notes 11 1 29–42.
IEEE C. Nur, “Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials”, Math. Sci. Appl. E-Notes, vol. 11, no. 1, pp. 29–42, 2023, doi: 10.36753/mathenot.1110497.
ISNAD Nur, Cemile. “Computing Eigenvalues of Sturm--Liouville Operators With a Family of Trigonometric Polynomial Potentials”. Mathematical Sciences and Applications E-Notes 11/1 (March 2023), 29-42. https://doi.org/10.36753/mathenot.1110497.
JAMA Nur C. Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials. Math. Sci. Appl. E-Notes. 2023;11:29–42.
MLA Nur, Cemile. “Computing Eigenvalues of Sturm--Liouville Operators With a Family of Trigonometric Polynomial Potentials”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 1, 2023, pp. 29-42, doi:10.36753/mathenot.1110497.
Vancouver Nur C. Computing Eigenvalues of Sturm--Liouville Operators with a Family of Trigonometric Polynomial Potentials. Math. Sci. Appl. E-Notes. 2023;11(1):29-42.

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