The inverse scattering problem of quantum theory

The inverse phase-type scattering problem for the boundary-value problem
\begin{gather} -y''+q(x)y=k^2y\quad(0\le x<\infty),\\ y'(0)=hy(0). \end{gather}
is studied.
It is shown that, for each function $\delta(k)$ satisfying the hypotheses of Levinson's theorem, there exists a problem (1)–(2) with $h\ne\infty$ and another problem (1)–(2) with $h=\infty$ (i.e., with the boundary condition $y(0)=0$).
The solvability condition for the Riemann-Hilbert problem is used more directly than has been done heretofore by others in deriving boundary condition (2).