The Levinson-Type Formula for a class of Sturm-Liouville Equation

Abstract

The boundary value problem
\[-{\psi}''+q(x)\psi={\lambda}^2 \psi, \quad 0<x<\infty,\]
\[{\psi}'(0)-(\alpha_{0}+\alpha_{1}\lambda){\psi}(0)=0 \]
is considered, where $\lambda$ is a spectral parameter, $ q(x) $ is real-valued function such that
\begin{equation*}
\int\limits_{0}^{\infty}(1+x)|q(x)|dx<\infty
\end{equation*}
with $\alpha_{0}, \alpha_{1}\geq0$ ( $\alpha_{0},\alpha_{1}\in \mathbb{R}$).

In this paper, for the above-mentioned boundary value problem, the scattering data is considered and the characteristics properties (such as continuity of the scattering function $ S(\lambda) $ and giving the Levinson-type formula) of this data are studied.
{\small \bf Keywords. }{Scattering data, scattering function, Gelfand-Levitan-Marchenko equation, Levinson-type formula.}