Could someone please tell me why this is true ?
Let $g=g(x,z)$ $f(x)=\exp(ikx)\left(1+i{g \over k}-{g_z \over k^2}\right)\bigg|_{z=x}-{1\over k^2}\int_x^\infty g_{zz}\exp(ikz)\,\,dz$where $g,g_z\to 0 $ as $z\to\infty$.
Also, let $g_{xx}=g_{zz}+ug$ when $z>x$ and where $u=-2(g_x+g_z)\big|_{z=x}$.
Then $\left\{-\partial_x^2-2(g_x+g_z)\big|_{z=x}\right\}f=k^2f$