On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition

Abstract

An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line 0, ∞ 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.

Characteristic properties of scattering data of a boundary value problem

“…Hence when E(0) = 0, K 1 (z) is bounded solution of the equation (20). Bounded solution of this equation is integrable on the half axis [a(1 − α), ∞).…”

“…According to Lemma 2.2 in [20], the equation E(λ) = 0 has only finite number of simple roots in the half plane λ > 0; moreover, these roots lie on the imaginary axis.…”

On a Boundary Value Problem for a Polynomial Pencil of the Sturm-Liouville Equation with Spectral Parameter in Boundary Conditions

“…There has been a growing interest in SturmLiouville problems with spectral parameter in boundary conditions in recent years and there are a lot of articles on this subject in the literature. For more detailed analysis we refer to the papers [1]- [9] and the references therein. In the case 1 n > the simple boundary value problem for the Equation (1) with conditions ( ) ( )…”

Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval

“…Inverse scattering problem for Equation (1) with ( ) 0 p x = on the half line [ ) 0, +∞ was investigated and the complete solution of this problem was given in [38] where the new integral representation, similar to transformation operators [9], was obtained for the Jost solution of the discontinuous Sturm-Liouville equation. Direct and inverse scattering problems on the half-line for the discontinuous Sturm-Liouville equation with eigenparameter dependent boundary conditions have been investigated in [41]. The direct and inverse spectral problem for the Equation (1) in the case ( ) 0 p x = with some separated boundary conditions on the interval ( ) 0, π recently has been investigated in [39] [40] [42], where the new integral representations for solutions have been also constructed.…”