Abstract
In the present paper, boundary value problems for discontinuous non-selfadjoint Sturm–Liouville operators on a finite interval with boundary conditions nonlinearly dependent on the spectral parameter are considered, and a new approach for studying the asymptotic representation of the eigenfunctions and their partial derivatives is presented. We obtain the asymptotic representation of the solutions and the eigenvalue, and study some of their main properties. Then, we provide a constructive procedure to obtain the asymptotic form of the eigenfunctions and their partial derivatives in discontinuous case by the canonical form of the Bessel functions \(J_{\frac{1}{2}}(z)\), \(J_{\frac{3}{2}}(z)\) and their derivatives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amirov, R.K.: On a system of Dirac differential equations with discontinuity conditions inside an interval. Ukr. Math. J. 57, 712–727 (2005)
Bailey, P.B., Everitt, W.N., Weidmann, J., Zettl, A.: Regular approximations of singular Sturm–Liouville problems. Results Math. 23, 3–22 (1993)
Buterin, S.A.: On inverse spectral problem for non-selfadjoint Sturm–Liouville operator on a finite interval. J. Math. Anal. Appl. 335, 739–749 (2007)
Eves, H.W.: Functions of a Complex Variable. Prindle, Weber and Schmidt Inc., Boston (1966)
Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and their Applications. NOVA Science Publishers, New York (2001)
Freiling, G., Yurko, V.A.: Inverse problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter. Inverse Probl. 26, 055003 (2010). https://doi.org/10.1088/0266-5611/26/5/055003
Gesztesy, F., Simon, B.: Uniqueness theorems in inverse spectral theory for one-dimensional Schr\(\ddot{o}\)dinger operators. Trans. Am. Math. Soc. 348, 349–373 (1996)
Gesztesy, F., Simon, B.: On local Borg–Marchenko uniqueness results. Commun. Math. Phys. 211, 273–287 (2000)
Goktas, S., Kerimov, N.B., Maris, E.A.: On the uniform convergence of spectral expansions for a spectral problem with a boundary condition rationally depending on the eigenparameter. J. Korean Math. Soc. 54, 1175–1187 (2017)
Guliyev, N.J.: Inverse eigenvalue problems for Sturm–Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl. 21, 1315–1330 (2005)
Hald, O.H.: Discontinuous inverse eigenvalue problems. Commun. Pure Appl. Math. 37, 539–577 (1984)
Harris, B.J., Marzano, F.: Eigenvalue approximations for linear periodic differential equations with a singularity. Electron. J. Qual. Theory Differ. Equ. 7, 1–18 (1999)
Koyunbakan, H., Gulsen, T., Yilmaz, E.: Inverse nodal problem for a \(p\)-Laplacian Sturm–Liouville equation with polynomially boundary condition. Electron. J. Differ. Equ. 7, 1–9 (2018)
Lapwood, F.R., Usami, T.: Free Oscillations of the Earth. Cambridge University Press, Cambridge (1981)
Levitan, B.M.: Inverse Sturm–Liouville Problems, Nauka, Moscow, 1984; English transl. VNU Sci. Press, Utrecht (1987)
Levitan, B.M., Gasymov, M.G.: Determination of a differential equation from two of its spectra. Russ. Math. Surv. 19, 1–63 (1964)
Levitan, B.M., Sargsyan, I.S.: Some problems in the theory of the Sturm–Liouville equation. Russ. Math. Surv. 15, 3–98 (1960)
Likov, A.V., Mikhailov, Y.A.: The Theory of Heat and Mass Transfer. Qosenergaizdat, Moscow-Leningrad (1963)
Litvinenko, O.N., Soshnikov, V.I.: The Theory of Heterogeneous Lines and Their Applications in Radio Engineering. Radio, Moscow (1964)
Marchenko, V.A.: Sturm–Liouville Operators and Their Applications, Revised edn. AMS Chelsea Publishing, Stanford (2011)
Mosazadeh, S.: Energy levels of a physical system and eigenvalues of an operator with a singular potential. Rep. Math. Phys. 82, 137–148 (2018)
Neamaty, A.: Some properties of the eigenfunctions of Sturm–Liouville problem with two turning points. Iran. J. Sci. Technol. Trans. A 27, 381–388 (2003)
Neamaty, A., Mosazadeh, S.: On the canonical solution of Sturm–Liouville problem with singularity and turning point of even order. Can. Math. Bull. 54, 506–518 (2011)
Olver, F.W.J., Maximon, L.C.: NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
Shkalikov, A.A.: Boundary-value problems for ordinary differential equations with a parameter in the boundary conditions. Funct. Anal. Appl. 16, 324–326 (1982)
Shkalikov, A.A.: Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Math. Sci. 33, 1311–1342 (1986)
Shkalikov, A.A.: Basis properties of root functions of differential operators with spectral parameter in the boundary conditions. Differ. Equ. 55, 631–643 (2019)
Trubowitz, E.: Inverse Spectral Theory. Academic Press Inc., London (1987)
Wang, Y.P., Koyunbakan, H.: On the Hochstadt–Lieberman theorem for discontinuous boundary-valued problems. Acta Math. Sin. Eng. Ser. 30, 985–992 (2014)
Acknowledgements
This research is partially supported by the University of Kashan under Grant No. 985969/2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mosazadeh, S. A new approach to asymptotic formulas for eigenfunctions of discontinuous non-selfadjoint Sturm–Liouville operators. J. Pseudo-Differ. Oper. Appl. 11, 1805–1820 (2020). https://doi.org/10.1007/s11868-020-00350-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-020-00350-2
Keywords
- Non-selfadjoint Sturm–Liouville operator
- Boundary conditions nonlinearly dependent on the spectral parameter
- Bessel functions
- Asymptotic representation
- Eigenfunctions