Let $q \in C^\infty(\mathbb{R}) \cap L^\infty(\mathbb{R})$ be an even function, with $q(0)<0$, and assume $q$ converges exponentially fast to $\bar{q}>0$ for $x \to \pm\infty$. Let $f \in L^2(\mathbb{R})$ be an odd function. Then, I am looking for a solution of $$ -u'' + q(x)u = f(x) $$ such that $u \in L^2(\mathbb{R})$.
The problem looks very innocent, reminiscent of a Sturm–Liouville problem (but on the whole of $\mathbb{R}$). However, the non-positivity of $q$ around $0$ jeopardizes many of my attempts to approach it. I also thought of going to the Fourier domain, but then I get stuck on an integral equation, as I am stuck on the above. Am I missing a reference where this problem is already extensively treated? Or otherwise, any suggestion on how to approach the existence of an $L^2$ solution?