Does anybody have any good references on the spectral theory of time-independent Schrödinger equations?
In particular, I'm looking at an equation of the form with,
$$ -y''(x)+q(x)y(x)=\lambda y(x) $$
on $(1,\infty)$ with a potential $q(x)$ that has the properties $q(1)=0$, $\lim_{x\rightarrow \infty}q(x)=C\in \mathbb{R}_+$, but in particular $q\not\in L^1(1,\infty)$. I would like to know under what conditions, I can infer the existence of a negative eigenvalue $\lambda$. A reference which discusses this situation would be ideal.