Spectral properties and inverse nodal problems for singular diffusion equation

Abstract

In this study, some properties for the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues were learned, then the solutions of the inverse problem were given to determine the potential function and parameters of the boundary condition with the help of a dense set of nodal points and lastly we obtain a constructive solution to the inverse problems of this class.

Keywords

• singular diffusion equation
• inverse spectral problems
• inverse nodal problems

References

1. [1] S. Albeverio, F. Gesztesy, R. Hoegh-Kron and H. Holden, Solvable models in quantum mechanics, Springer, New York-Berlin, 1988.
2. [2] R.Kh. Amirov and I.M. Guseinov, Boundary value problems for a class of Sturm- Liouville operator with nonintegrable potential, Dif. Eq. 38(8), 1195-1197, 2002.
3. [3] S.A. Buterin and S.T. Chung, Inverse nodal problem for differential pencils, Appl. Math. Letters 22, 1240-1247, 2009.
4. [4] I.M. Guseinov and L.I. Mammadova, Properties of the eigenvalues of the Sturm- Liouville operator with discontinuity conditions inside the interval, Pross. Baku State University, Phys-Math. Sci. Series 3, 2011.
5. [5] I.M. Guseinov and L.I. Mammadova, Reconstruction of the diffusion equation with singular coefficients for two spectra, Doklady Math. 90(1), 401-404, 2014.
6. [6] B.Y. Levin, Lectures on entire functions, Transl. Math. Monographs 150, Amer. Math. Soc, Providence RI, 1996.
7. [7] M.Dzh. Manafov, Inverse spectral and inverse nodal problems for Sturm-Liouville equations with point $\delta$ and $\delta^{^{\prime}}$- interactions, Proc. of the Institute of Math. and Mech. 45(2), National Acad. Sci. Azerbaijan, 286-294, 2019.
8. [8] V.A. Marchenko, Sturm–Liouville operators and their applications, Naukova Dumka, Kiev, 1977. English transl., Birkhäuser, Basel, 1986.

9. [9] A.A. Nabiev and R.Kh. Amirov, Integral representations for the solutions of the generalized Schroedinger equation in a finite interval, Adv. Pure Math. 5(13), 777-795, 2015.
10. [10] N. Pronska, Reconstruction of energy-dependent Sturm-Liouville equations from two spectra, Integral Equations Operator Theory 76, 403-419, 2013.
11. [11] A.M. Savchuk, On the eigenvalues and eingenfunctions of the Sturm-Liouville Operator with singular potential, Math. Notes 69(2), 245-252, 2001.
12. [12] C.F. Yang, An inverse problem for a differential pencil using nodal points as data, Israel J. Math 204, 431-446, 2014.
13. [13] Ch.G. Ibadzadeh, L.I. Mammadova and I.M. Nabiev, Inverse problem of spectral analysis for diffusion operator with nonseparated boundary conditions and spectral parameter in boundary condition, Azerbaijan J. Math. 9(1), 171-189, 2019.
14. [14] I.M. Nabiev, Reconstruction of the differential operator with spectral parameter in the boundary condition, Mediterr. J. Math. 19(3), 1-14, 2022.
15. [15] L.I. Mammadova, I.M. Nabiev and Ch.H. Rzayeva, Uniqueness of the solution of the inverse problem for differential operator with semiseparated boundary conditions, Baku Math. J. 1(1), 47-52, 2022.
16. [16] N.J. Guliyev, Inverse square singularities and eigenparameter-dependent boundary conditions are two sides of the same coin, The Quarterly J. Math. 74(3), 889-910, 2023.
17. [17] N.J. Guliyev, Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter, J. Math. Phys. 60(6), 063501, 23 pp, 2019.
18. [18] N.J. Guliyev, Essentially isospectral transformations and their applications, Ann. Mat. Pura Appl. 199(4), 1621-1648, 2020.
19. [19] N.J. Guliyev, On two-spectra inverse problems, Proc. Amer. Math. Soc. 148(10), 4491-4502, 2020.
20. [20] Y.H. Cheng, Reconstruction and stability of inverse nodal problems for energydependent p-Laplacian equation, J. Math. Ann. Appl. 491(2), 124388, 2020.
21. [21] C. F. Yang, Direct and inverse nodal problem for differential pencil with coupled boundary conditions, Inverse Prob. Sci. Eng. 21(4), 562-584, 2013.
22. [22] M. Dzh. Manafov and A. Kablan, Inverse spectral and inverse nodal problems for energydependent Sturm-Liouville equations with -interaction, Elec. J. Dif. Eq. 2015(26), 1-10, 2015.