Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 171967 17 pages doi 2010 171967 Research Article On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition Khanlar R. Mamedov Mathematics Department Science and Letters Faculty Mersin University 33343 Mersin Turkey Correspondence should be addressed to Khanlar R. Mamedov hanlar@ Received 9 April 2010 Accepted 22 May 2010 Academic Editor Michel C. Chipot Copyright 2010 Khanlar R. Mamedov. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line 0 to 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. 1. Introduction We consider inverse scattering problem for the equation -p q x y k2p x y 0 x to with the boundary condition - a p 0 - 2p 0 A2 1p 0 - ft p 0 where A is a spectral parameter q x is a real-valued function satisfying the condition to 1 x q x dx to 0 p x is a positive piecewise-constant function with a finite number of points of discontinuity ữi pi i 1 2 are real numbers and Y ữ1p2 - X2Ộ1 0. 2 Boundary Value Problems The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line 0 to for the boundary value problem - . In the case p x 1 the inverse problem of scattering theory for with boundary condition not containing spectral parameter was completely solved by Marchenko 1 2 Levitan 3 4 Aktosun 5 as well as .
