Spectral Properties of the Sturm-Liouville Operator Produced by the Unseparated Boundary Conditions with Spectral Parameter

Abstract

In this study, firstly, the basic properties of the spectrum of the investigated problem were learned, sine and cosine type solutions were defined, their behaviors were examined and the properties of the solution of the given problem were learned with their help. Next, the characteristic equation of the studied problem was formed with the help of sine and cosine type solutions. Using the characteristic equation, the asymptotic behavior of the eigenvalues of the given problem and the ordering of the eigenvalues of the boundary value problems $L(\alpha _{j}),$ $j=1,$ $2$ when $\alpha_{1} $ $<\alpha _{2}$ are learned.

Keywords

• Eigenvalue
• Eigenfunction
• Unseparated boundary conditions

References

1. [1] Amirov, R.Kh., Cakmak, Y., Inverse problem for Sturm-Liouville operator with respect to a spectrum and normalizing numbers, Cumhuriyet Journal of Science, 24(1)(2003), 34–50.
2. [2] Amirov, R.Kh., Keskin, B., Ozkan, A.S., Direct and inverse problems for the impulsive Sturm-Liouville boundary value problem where boundary conditions include the spectral parameter, Cumhuriyet Journal of Science, 27(2)(2006), 13–23.
3. [3] Amirov, R., Durak, S., Behaviors of eigenvalues and eigenfunctions of the singular Shrödinger operator, Turkish Journal of Mathematics and Computer Science, 12(2)(2020), 151–156.
4. [4] Gasymov, M.G., Guseinov, I.M., Nabiev, I.M., An inverse problem for the Sturm-Liouville operator with nonseparable self-adjoint boundary conditions, Siberian Mathematical Journal, 31(6)(1990), 910–918.
5. [5] Gasymov, M.G., Guseinov, G. Sh., Reconstruction of a diffusion operator from the spectral data, Dokl. Akad.Nauk. Azerb. SSR, 37(2)(1981), 19–23.
6. [6] Guliyev, N.J., Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse Problems, IOP Publishing, 21(4)(2005), 1315–1330.
7. [7] Guo, Y.,Wei, G., On the reconstruction of the Sturm-Liouville problems with spectral parameter in the discontinuity conditions, Results. Math., 65(2014), 385–398.
8. [8] Guseinov, I.M., Nabiev, I.M., On one class of inverse boundary-value problems for the Strum-Liouville operators, Differents. Uravn., 25(7)(1989), 1114–1120.

9. [9] Guseinov, I.M., Nabiev, I.M., Solution of a class of inverse boundary-value Sturm-Liouville problems, Sbornik: Mathematics, 186(5)(1995), 661–674.
10. [10] Güldü, Y., Amirov, R.K., Topsakal, N., On impulsive Sturm-Liouville operators with singularity and spectral parameter in boundary conditions, Ukrainian Mathematical Journal, 64(2013), 1816–1838.
11. [11] Ibadzadeh, Ch.G., Nabiyev, I.M., An inverse problem for Sturm-Liouville operator with non-separated boundary conditions containing the spectral parameter, Journal of Inverse and III Posed Problems, 24(4)(2016), 407–411.
12. [12] Nabiev, I.M., Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators, Mathematical Notes, 67(3)(2000), 309–319.
13. [13] Nabiev, I.M., The uniqueness of reconstruction of quadratic bundle for Sturm-Liouville operators, Proceedings of IMM of NAS of Azerbaijan, 23(2004), 91–96.
14. [14] Nabiev, I.M., Shukurov, A.Sh., Properties of the spectrum and uniqueness of reconstruction of Sturm-Liouville operator with a spectral parameter in the boundary condition, Proc. of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 40, Special Issue, (2014), 332–341.
15. [15] Marchenko, V.A., Sturm-Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977, English transl.: Birkhauser, 1986.
16. [16] Yurko, V.A., The inverse spectral problem for differential operators with non-separated boundary conditions, Journal of Inverse and III Posed Problems, 28(4)(2020), 567–616.