The forward problem is a second order Sturm-Liouville operator
$ - \frac{d^{2}}{dx^{2}}y(x)+q(x)y(x)=zy(x) $
with the boundary conditions $ y(0)=0=y(\infty) $.
If I know the spectral measure function $ \sigma (x) =\sum_{\lambda_{n} \le x}1 $, then can I reconstruct the inverse of the potential $ q^{-1}(x) $?
My question is, how do I use the Gelfand-Levitan-Marchenko theory to reconstruct the potential $ q(x) $?