Spectral Characteristics of the Sturm-Liouville Problem with Spectral Parameter-Dependent Boundary Conditions

Abstract

We consider the Sturm-Liouville problem on the half line $(0 \leq x<\infty)$, where the boundary conditions contain polynomials of the spectral parameter. We define the scattering function and present the spectrum of the boundary value problem. The continuity of the scattering function is discussed. In a special case, the Levinson-type formula is introduced, demonstrating that the increment of the scattering function's logarithm is related to the number of eigenvalues.

Keywords

• Levinson-type formula
• scattering function
• spectral parameter dependent boundary condition

References

1. D. S. Cohen, An integral transform associated with boundary conditions containing an eigenvalue parameter, SIAM Journal on Applied Mathematics 14 (5) (1966) 1164-1175.
2. L. Collatz, Eigenwertaufgaben mit technischen anwendungen, Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1949.
3. P. A. Binding, P. J. Browne, B. A. Watson, Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Journal of the London Mathematical Society 62 (1) (2000) 161-182.
4. P. A. Binding, P. J. Browne, B. A. Watson, Sturm Liouville problems with boundary conditions rationally dependent on the eigenparameter, II, Journal of Computational and Applied Mathematics 148 (1) (2002) 147-168.
5. C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 77 (3-4) (1977) 293-308.
6. Ch. G. Ibadzadeh, L. I. Mammadova, I. M. Nabiev, Inverse problem of spectral analysis for diffusion operator with nonseparated boundary conditions and spectral parameter in boundary condition, Azerbaijan Journal of Mathematics 9 (1) (2019) 171-189.
7. I. M. Nabiev, Reconstruction of the differential operator with spectral parameter in the boundary condition, Mediterranean Journal of Mathematics 19 (3) (2022) 1-14.
8. L. I. Mammadova, I. M. Nabiev, Spectral properties of the Sturm–Liouville operator with a spectral parameter quadratically included in the boundary condition, Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp'yuternye Nauki 30 (2) (2020) 237–-248.

9. A. A. Nabiev, On a boundary value problem for a polynomial pencil of the Sturm-Liouville equation with spectral parameter in boundary conditions, Applied Mathematics 7 (18) (2016) 2418-2423.
10. V. N. Pivovarchik, Direct and inverse problems for a damped string, Journal of Operator Theory 42 (1999) 189-220.
11. A. Çöl, Inverse spectral problem for Sturm-Liouville operator with discontinuous coefficient and cubic polynomials of spectral parameter in boundary condition, Advances in Difference Equations 2015 (2015) 1-12.
12. Kh. R. Mamedov, Uniqueness of the solution to the inverse problem of scattering theory for the Sturm–Liouville operator with a spectral parameter in the boundary condition, Mathematical Notes 74 (2003) 136-140.
13. Kh. R. Mamedov, F. A. Cetinkaya, Boundary value problem for a Sturm-Liouville operator with piecewise continuous coefficient, Hacettepe Journal of Mathematics and Statistics 44 (4) (2015) 867-874.
14. Kh. R. Mamedov, H. Menken, On the inverse problem of scattering theory for a differential operator of the second order, North-Holland Mathematics Studies 197 (2004) 185-194.
15. D. Bolle, Sum rules in scattering theory and applications to statistical mechanics, Mathematics + Physics, Lectures on Recent Results 2 (1986) 84-153.
16. Z. Q. Ma, The Levinson theorem, Journal of Physics A: Mathematical and General 39 (48) (2006) R625.
17. R. G. Newton, Scattering theory of waves and particles, Springer-Verlag, New York, 1982.
18. N. Levinson, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Danske Videnskab Selskab Matematisk-Fysiske Meddelelser 25 (9) (1949) 29.
19. V. A. Marchenko, Sturm–Liouville operators and applications, Birkhäuser Verlag, Basel, 1986.
20. S. Goktas, Kh. R. Mamedov, The Levinson-type formula for a class of Sturm-Liouville equation, Facta Universitatis, Series: Mathematics and Informatics 35 (4) (2020) 1219-1229.
21. Kh. R. Mamedov, N. P. Kosar, Continuity of the scattering function and Levinson type formula of a boundary-value problem, International Journal of Contemporary Mathematical Sciences 5 (4) (2010) 159-170.
22. Ö. Mızrak, Kh. R. Mamedov, A. M. Akhtyamov, Characteristic properties of scattering data of a boundary value problem, Filomat 31 (12) (2017) 3945-3951.