Characteristic properties of scattering data of a boundary value problem

Abstract

Consider the differential equation ?y''+ q(x)y = ?2?(x)y, 0 < x < ? (1) with boundary condition -(?1y(0)- ?2y'(0)) = ?2(?1y(0)-?2y'(0)). (2) Here q(x) is a real valued function such that ??0 (1 + x)?q(x)?dx < ? and ?(x) is a real valued piecewise continuous function. It is known that the boundary value problem (3)-(4) has only finite number of simple negative eigenvalues -?21,...,-?2n, (?j > 0) and the half axis constitutes absolutely continuous spectrum. For normalized eigenfunctions of the problem (3)-(4) we have the asymptotic formulae as x ? ? uj(x)~ mje-?jx j = 1,..., n, u(?,x)~ e-i?x - S(?)ei?x, -? < ? < 1. So at infinity behaviour of the radial waves is defined by {S(?)(-? < ? < 1),?2k, mk (k=1...n)}. These are called scattering data of the (3)-(4) boundary value problem. In this work characteristic properties of the scattering data will be investigated.