Consider the Sturm-Liouville problem
$$-u''(x) + V(x)u(x) = \lambda u(x)$$
for $u : \mathbb R \rightarrow \mathbb R$, $\lambda \in \mathbb R$. I am looking for a method to find the eigenvalues $\lambda$ which produce solutions $u \in L^2(\mathbb R)$ for various choices of $V$, e.g. $V(x) = c/\cosh^2(Cx)$ or $V(x) = e^x$. Is there a general procedure for doing this? If not, can someone provide suggestions for finding the eigenvalues at least in the two cases I mentioned?