Characteristic Properties of Scattering Data of a Boundary Value Problem

Abstract

Consider the differential equation
\begin{equation}
-y''+q(x)y=\lambda^2\rho(x)y, \quad 0<x<\infty \label{eqn}
\end{equation}
with boundary condition
\begin{equation}
-(\alpha_1 y(0)-\alpha_2 y'(0))=\lambda^2(\beta_1y(0)-\beta_2y'(0)). \label{bdv}
\end{equation} Here $q(x)$ is a real valued function such that $$\int_0^\infty (1+x)|q(x)|dx<\infty$$ and $\rho(x)$ is a real valued piecewise continuous function. It is known that the boundary value problem (\ref{eqn})-(\ref{bdv}) has only finite number of simple negative eigenvalues $-\mu_1^2,\cdots,-\mu_n^2,(\mu_j>0)$ and the half axis constitutes absolutely continuous spectrum. For normalized eigenfunctions of the problem (\ref{eqn})-(\ref{bdv}) we have the asymptotic formulae as $x\rightarrow\infty$
\begin{align}
u_j(x)&\sim m_j e^{-\mu_jx},\qquad j=1,\dots,n ,\nonumber \\
u(\lambda,x)& \sim e^{-i\lambda x}-S(\lambda)e^{i\lambda x}, \qquad -\infty<\lambda<\infty. \nonumber
\end{align}
So at infinity behaviour of the radial waves is defined by $\{ S(\lambda) \ (-\infty<\lambda<\infty), -\mu_k^2,\ m_k \ (k=1\dots n) \}$. These are called scattering data of the (\ref{eqn})-(\ref{bdv}) boundary value problem. In this work characteristic properties of the scattering data will be investigated.