On the Jost Solutions of A Class of the Quadratic Pencil of the Sturm-Liouville Equation
Year 2023,
Volume: 18 Issue: 1, 18 - 27, 25.05.2023
Anar Adiloğlu
,
Döndü Nurten Cücen
Abstract
In this study we construct new integral representations of Jost-type solutions of the quadratic pencil of the Sturm-Liouville equation with the piece-wise constant coefficient on the entire real line. Our aim is to express the special solutions of the Sturm-Liouville quadratic pencil in the form of some integral operators which kernels is related with the potential function of the Sturm-Liouville equation. This problem is technically diffucult due to the discontinous coefficient which causes the kernel function to also have a jump discontinuity.
References
- B. M. Levitan and M. G. Gasymov, “Determination of a differential equation by two spectra”, Uspehi Mat. Nauk, 19(2), 3-63, 1964.
- M. G. Gasymov, “The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient”, in Non- classical methods in geophysics, Editors: M. M. Lavrent'ev. Novosibirsk Nauka, 1977, 37-44 (in Russian).
- F. G. Maksudov and G. Sh. Guseinov, “On the solution of the inverse scattering problem for the quadratic pencil of the Schrödinger equation on the full-line”, Dokl. Akad. Nauk USSR, 289(1), 42-46, 1986.
- H. M. Huseynov and J. A. Osmanl, “Inverse scattering problem for one-dimensional Schrö dinger equation with discontinuity conditions”, Journal of Mathematical Physics, Analysis, Geometry, 9(3), 332-359, 2013.
- A. A. Nabiev and Kh. R. Mamedov, “On the Jost solutions for a class of Schrödinger equations with piecewise constant coefficients”, Journal of Mathematical Physics, Analysis, Geometry, 11, 279-296, 2015.
- M. Jaulent and C. Jean, “The inverse problem for the one dimensional Schrödinger equation with an energy dependent potential, I, II”, Ann. Inst. Henri Poincare, 25, 105-118, 1976.
- D. J. Kaup, “A higher-order water-wave equation and the method for solving it”, Prog. Theor. Phys., 54, 396-408, 1975.
- V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Basel: Birkhauser, 1986.
- Kh. R. Mamedov, “On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition”, Boundary Value Problems, Article ID 171967, 17 pages, 2010.
- Kh. R. Mamedov and A. A. Nabiev, “Inverse problem of scattering theory for a class one-dimensional Schrö dinger equation”, Quastiones Math., 42(7), 841-856, 2019.
- Y. Kamimura, “An inversion formula in energy dependent scattering”, Journal of Integral Equations and Applications, 19(4), 473-512, 2007.
- K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1989.
- A. A. Nabiev, “Direct and inverse scattering problem for the one dimensional Schrödinger equation with energy dependent potential and discontinuity conditions”, Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 40, 315-331, 2014.
Year 2023,
Volume: 18 Issue: 1, 18 - 27, 25.05.2023
Anar Adiloğlu
,
Döndü Nurten Cücen
References
- B. M. Levitan and M. G. Gasymov, “Determination of a differential equation by two spectra”, Uspehi Mat. Nauk, 19(2), 3-63, 1964.
- M. G. Gasymov, “The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient”, in Non- classical methods in geophysics, Editors: M. M. Lavrent'ev. Novosibirsk Nauka, 1977, 37-44 (in Russian).
- F. G. Maksudov and G. Sh. Guseinov, “On the solution of the inverse scattering problem for the quadratic pencil of the Schrödinger equation on the full-line”, Dokl. Akad. Nauk USSR, 289(1), 42-46, 1986.
- H. M. Huseynov and J. A. Osmanl, “Inverse scattering problem for one-dimensional Schrö dinger equation with discontinuity conditions”, Journal of Mathematical Physics, Analysis, Geometry, 9(3), 332-359, 2013.
- A. A. Nabiev and Kh. R. Mamedov, “On the Jost solutions for a class of Schrödinger equations with piecewise constant coefficients”, Journal of Mathematical Physics, Analysis, Geometry, 11, 279-296, 2015.
- M. Jaulent and C. Jean, “The inverse problem for the one dimensional Schrödinger equation with an energy dependent potential, I, II”, Ann. Inst. Henri Poincare, 25, 105-118, 1976.
- D. J. Kaup, “A higher-order water-wave equation and the method for solving it”, Prog. Theor. Phys., 54, 396-408, 1975.
- V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Basel: Birkhauser, 1986.
- Kh. R. Mamedov, “On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition”, Boundary Value Problems, Article ID 171967, 17 pages, 2010.
- Kh. R. Mamedov and A. A. Nabiev, “Inverse problem of scattering theory for a class one-dimensional Schrö dinger equation”, Quastiones Math., 42(7), 841-856, 2019.
- Y. Kamimura, “An inversion formula in energy dependent scattering”, Journal of Integral Equations and Applications, 19(4), 473-512, 2007.
- K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, New York, 1989.
- A. A. Nabiev, “Direct and inverse scattering problem for the one dimensional Schrödinger equation with energy dependent potential and discontinuity conditions”, Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 40, 315-331, 2014.