Abstract
We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with periodic and antiperiodic boundary conditions for special potentials that are trigonometric polynomials. Moreover, we give error estimations and, finally, we present an example.
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References
O. A. Veliev, “Isospectral Mathieu–Hill operators,” Lett. Math. Phys. 103, 919–925 (2013).
B. M. Brown, M. S. P. Eastham, and K. M. Schmidt, Periodic Differential Operators, Operator Theory: Advances and Applications, (Birkhäuser/Springer: Basel AG, Basel, 2013).
D. M. Levy and J. B. Keller, “Instability intervals of Hill’s equation,” Comm. Pure Appl. Math 16, 469–476 (1963).
W. Magnus and S. Winkler, Hill Equation (Interscience Publishers, John Wiley, 1969).
V. Marchenko, Sturm–Liouville Operators and Applications (Birkhäuser Verlag, Basel, 1986).
M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic Press: Edinburg, UK, 1973).
N. Dernek and O. A. Veliev, “On the Riesz basisness of the root functions of the nonself-adjoint Sturm–Liouville operators,” Israel Journal of Mathematics 145, 113–123 (2005).
N. B. Kerimov and K. R. Mamedov, “On the Riesz basis property of the root functions in certain regular boundary value problems,” Math. Notes 64 (4), 483–487 (1998).
A. A. Shkalikov and O. A. Veliev, “On the Riesz basis property of the eigen- and associated functions of periodic and antiperiodic Sturm–Liouville problems,” Math. Notes 85 (5), 647–660 (2009).
A. L. Andrew, “Correction of finite element eigenvalues for problems with natural or periodic boundary conditions,” BIT 28 (2), 254–269 (1988).
A. L. Andrew, “Correction of finite difference eigenvalues of periodic Sturm–Liouville problems,” Journal of Australian Mathematical Society B. 30 (4), 460–469 (1989).
D. J. Condon, “Corrected finite difference eigenvalues of periodic Sturm–Liouville problems,” Applied Numerical Mathematics 30 (4), 393–401 (1999).
G. Vanden Berghe, M. Van Daele, and H. De Meyer, “A modified difference scheme for periodic and semiperiodic Sturm–Liouville problems,” Applied Numerical Mathematics 18 (1–3), 69–78 (1995).
X. Ji and Y. S. Wong, “Prüfer method for periodic and semiperiodic Sturm–Liouville eigenvalue problems,” International Journal of Computer Mathematics 39, 109–123 (1991).
X. Z. Ji, “On a shooting algorithm for Sturm–Liouville eigenvalue problems with periodic and semi-periodic boundary conditions,” Journal of Computational Physics 111 (1), 74–80 (1994).
Y. S. Wong and X. Z. Ji, “On shooting algorithm for Sturm–Liouville eigenvalue problems with periodic and semi-periodic boundary conditions,” Applied Mathematics and Computation 51 (2–3), 87–104 (1992).
V. Malathi, M. B. Suleiman, and B. B. Taib, “Computing eigenvalues of periodic Sturm–Liouville problems using shooting technique and direct integration method,” International Journal of Computer Mathematics 68 (1–2), 119–132 (1998).
S. Dinibütün Asymptotic and Numerical Analysis of the Non-Self-Adjoint differential Operator, PhD thesis, 2013.
S. Dinibütün and O. A. Veliev, “On the estimations of the small periodic eigenvalues,” Abstract and Applied Analysis, Article ID 145967 (2013).
C. Nur, “On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions,” Boundary Value Problems 2018:190 (2018).
M. G. Gasymov, “Spectral analysis of a class of second-order non-self-adjoint differential operators,” Funktsional. Anal. Prilozhen 14, 14–19 (1980).
N. B. Kerimov, “On a boundary value problem of N. I. Ionkin type,” Differ. Equ. 49 (10), 1233–1245 (2013).
O. A. Veliev, Spectral Analysis of the Non-Self-Adjoint Mathieu–Hill Operator, arXiv: https:// arxiv.org/abs/1202.4735v1 (2012).
O. A. Veliev and M. T. Duman, “The spectral expansion for a nonself-adjoint Hill operators with a locally integrable potential,” Journal of Mathematical Analysis and Applications 265 (1), 76–90 (2002).
G. D. Birkhoff, “Boundary value problems and expansion problem of ordinary linear differential equations,” Trans. Amer. Math. Soc. 9, 373–395 (1908).
N. Dunford and J. T. Schwartz, Linear Operators, Part 3, Spectral Operators (Wiley-Interscience, New York, 1988).
M. A. Naimark, Linear Differential Operators (George G. Harap & Company, 1967).
Y. D. Tamarkin, “Some general problem of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions,” Math. Zeit. 27, 1–54 (1927).
J. Pöschel and E. Trubowitz, Inverse Spectral Theory (Academic Press: Boston, Mass, USA, 1987).
C. Nur and O. A. Veliev, “On the estimations of the small eigenvalues of non-self-adjoint Sturm–Liouville operators,” TWMS J. App. Eng. Math. 9 (4), 882–893 (2019).
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The author is grateful for the Research Fund of Yalova University (project no. 2019/AP/0010).
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Nur, C. On the Estimates of Periodic Eigenvalues of Sturm–Liouville Operators with Trigonometric Polynomial Potentials. Math Notes 109, 794–807 (2021). https://doi.org/10.1134/S0001434621050114
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DOI: https://doi.org/10.1134/S0001434621050114