Eigenvalues of a Sturm-Liouville problem 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? 
 A: The eigenvalues $\lambda$ emerge from boundary conditions that are part of the differential operator.  In your case, $x \in (-\infty, \infty)$, so you may specify the behavior of $u$ as $x \rightarrow \infty$.  One way to attack this is to consider the Fourier transform of $u$:
$$ \hat{u}(k) = \int_{-\infty}^{\infty} dx \: u(x) \exp{(i 2 \pi k x)} $$
Applying the integral operator to the S-L equation, and integrating by parts (and assuming that $u$ and its derivative vanish at $\infty$), we get an integral equation for $ \hat{u}$:
$$ \int_{-\infty}^{\infty} dk' \: \hat{V}(k-k') \hat{u}(k') = (\lambda + 4 \pi^2 k^2) \hat{u}(k)  $$
where
$$ \hat{V}(k) = \int_{-\infty}^{\infty} dx \: V(x) \exp{(i 2 \pi k x)} $$
and we recover the solution from
$$ u(x) = \int_{-\infty}^{\infty} dk \: \hat{u}(k) \exp{(-i 2 \pi k x)} $$
A: Numerically the problem can be solved e.g. using the Numerov algorithm with the shooting method. There are only a few choices for $V$ for which there is an explicit analytic solution, some of them you can find here. 
One of it is $$V(x) = \frac{c}{\cosh^2(C x)}$$ which is called the Pöschl–Teller potential. However, it only has solutions in $L^2$ if $c<0$. After proper rescaling, you will find that the eigensolutions solutions are Legendre functions.
When $V(x) =e^{x}$ I believe that the eigenvalue problem does not have any solution in $L^2$.
A: There is a technique called "AIM" which stands for "asymptotic iteration method". See for instance this reference 1 and reference 2.
