I'm trying to find the eigenvalues for a Schrödinger's equation in a potential that looks like
$$ V(\ell) = \left(\alpha+\frac{1}{2}\right)^2 + \left(\left(\alpha e^{-\ell}-(\alpha+1)\right)^2 - (\alpha+1)^2\right).$$
So that I'm trying to solve something like
$$ -f''(\ell)+V(\ell)f(\ell) = Ef(\ell) $$
After substituting $r=e^{-\ell}$, my eigenvalue equation looks like
$$-r^2\Psi''(r)-r\Psi'(r) + \left[\left(\alpha r-(\alpha+1)\right)^2 - (\alpha+1)^2 + \left(\alpha+\frac{1}{2}\right)^2 - E \right]\Psi(r)=0 $$
which looks quite a lot like a Bessel equation, but I'm kind of lost after this. I tried substituting $x =\alpha r - (\alpha+1)$, but then I'd lose the Bessel-like derivatives on the first two terms.
I am chiefly interested in energies $E<\left(\alpha+1/2\right)^2$. I know that the first eigenvalue is $E=0$ and I know the solution, and I am really only trying to find the gap with the second eigenvalue.
I would appreciate any hints into what I can do to solve this.