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

Year 2023, Volume: 11 Issue: 1, 29 - 42, 28.03.2023

Cemile Nur

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.

Keywords

Eigenvalue estimations , Periodic and antiperiodic boundary conditions , Trigonometric polynomial potentials

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).