Abstract
An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line with a linear spectral parameter in the boundary condition. The scattering data of the problem are defined and a new fundamental equation is derived, which is different from the classical Marchenko equation. With help of this fundamental equation, in terms of the scattering data, the potential is recovered uniquely.
Highlights
We consider inverse scattering problem for the equation−ψ q x ψ λ2ρ x ψ 0 < x < ∞, 1.1 with the boundary condition− α1ψ 0 − α2ψ 0 λ2 β1ψ 0 − β2ψ 0, 1.2 where λ is a spectral parameter, q x is a real-valued function satisfying the condition ∞1 x q x dx < ∞, 1.3 ρ x is a positive piecewise-constant function with a finite number of points of discontinuity, αi, βi i 1, 2 are real numbers, and γ α1β2 − α2β1 > 0.Boundary Value ProblemsThe aim of the present paper is to investigate the direct and inverse scattering problem on the half-line 0, ∞ for the boundary value problem 1.1 – 1.3
In the case ρ x ≡ 1, the inverse problem of scattering theory for 1.1 with boundary condition not containing spectral parameter was completely solved by Marchenko 1, 2, Levitan 3, 4, Aktosun 5, as well as Aktosun and Weder 6
Solution of inverse scattering problem on the half-line 0, ∞ by using the transformation operator was reduced to solution of two inverse problems on the intervals 0, a and a, ∞
Summary
− α1ψ 0 − α2ψ 0 λ2 β1ψ 0 − β2ψ 0 , 1.2 where λ is a spectral parameter, q x is a real-valued function satisfying the condition. It turns out that in this case the discontinuity of the function ρ x strongly influences the structure of representation of the Jost solution and the fundamental equation of the inverse problem. When ρ x ≡ 1 in 1.1 with the spectral parameter appearing in the boundary conditions, the inverse problem on the half-line was considered by Pocheykina-Fedotova 15 according to spectral function, by Yurko 16–18 according to Weyl function, and according to scattering data in 19, 20. This type of boundary condition arises from a varied assortment of physical problems and other applied problems such as the study of heat conduction by Cohen 21 and wave equation by Yurko 16, 17. By substituting the expressions for the functions f x, λ and f x, λ in 1.1 , it can be shown that 1.11 holds
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