let be the differential equation
$$ -y''(x)+q(x)y(x)=0 $$ (1)
from the Cauchy method to solve differenial equations
$ -y''(x)+y(x)=f(x) $ so $ y(s)= \int_{0}^{s}dt \sin (s-t)f(s) $
then for the equation (1) is it valid the integral equation representation
$$ y(s)=\int_{0}^{s}dt \sin (s-t)q(s)y(s) $$
if we imposoe the asymptotic condtion $ y(x)=e^{iax} $as $ x \to \infty $ then the integral equation becomes (approximately)
$$ e^{ias} \approx \int_{0}^{s}dt \sin (s-t)q(x)y(s) $ $$
