Consider the Sturm Liouville equation:
$$ - \Psi_{xx} + u \Psi = \lambda \Psi $$
Given that the potential $u(x)$ decreases faster than $|x|^{-1- \epsilon}, \epsilon > 0$, when $x \rightarrow \pm \infty$ how can I show that there exists a Jost solution $f(x,k)$, which is uniquely fixed by its asymptotic behaviour:
$$ f(x,k) = e^{ikx} + o(1), x \rightarrow \infty $$
$$ f(x,k) \sim a(k)e^{ikx} - \overline{b(k)}e^{-ikx}, x \rightarrow - \infty $$