OK, I know the title is fundamentally wrong! But I guess you know where I'm going with it!
Basically, I'm wondering if it's possible to prove that if $\int_{x\in\Omega}G(x,x')h(x')dV=0$ for any $\Omega$, which includes the support of a localized potential, then $h(x)=0$ inside the domain $\Omega$.
I noticed that the fundamental Lemma of calculus of variation would do the trick, if $G(x,x')$ has some certain smoothness. But the problem is, $G(x,x')$ is the Green's function and it's singular at $x=x'$.
Any idea how I can get around this singularity? Or is there any contradicting example that could prove $h(x)$ is not necessarily zero?
If it helps, $G(\mathbf x,\mathbf x')=\int\frac{\phi_{k'}^*(\mathbf x')\phi_{k'}(\mathbf x)}{E_{k'}-E_k}d^3k'$ is the Green's function for periodic Schrodinger's equation.