Existence of nontrivial bounded eigenfunction to second-order linear ODE

Question

I am reading a paper, which concerns a second-order linear ODE of the form $$cu'-u''+f(x)u=0.$$ The corresponding linear operator is denoted by $\mathcal{L}_0$, so $$\mathcal{L}_0\phi=c\phi'-\phi''+f(x)\phi.$$ I know that there exists a positive solution $\phi_0(x)$ to this equation, which is in $H^2(\mathbb{R})$. So this function $\phi_0$ is a positive eigenfunction for $\mathcal{L}_0$ with eigenvalue $0$. The author claims that because the positivity of $\phi_0$ means we can use Liouville's theorem to show that the equation $$(\mathcal{L}_0+\delta)\phi=0$$ does not have a nontrivial bounded solution for any $\delta>0$. First, I have never heard of Liouville's theorem (except the one from complex analysis, but I doubt that one is meant). Second, I figured this might have something to do with this being a Sturm-Liouville type equation, since that would imply that $0$ is the smallest eigenvalue (as $\phi_0$ does not have any zeroes). However, we do not have any boundary conditions, so I fail to see how to apply Sturm-Liouville theory in this case. Can someone figure out what is going on here? Or provide an alternative proof? Thanks in advance.