Perturbation of Laplacian

Question

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; so for $f\in D(-\Delta)=H^2(\mathbb{R}^3)$ we have $$\Vert Vf\Vert_{L^2(\mathbb{R}^3)}\leq\Vert V\Vert_{L^3(\mathbb{R}^3)}\Vert f\Vert_{L^6(\mathbb{R}^3)}$$ Now I can use Gagliardo-Niremberg-Sobolev inequality to get $$\Vert f\Vert_{L^6(\mathbb{R}^3)}\leq C\Vert\Delta f\Vert_{L^2(\mathbb{R}^3)}^{\frac{1}{2}}\Vert f\Vert_{L^2(\mathbb{R}^3)}^{\frac{1}{2}}$$ Now I can use Young inequality $$\Vert f\Vert_{L^6(\mathbb{R}^3)}\leq C(\epsilon\Vert\Delta f\Vert_{L^2(\mathbb{R}^3)}+\frac{1}{\epsilon}\Vert f\Vert_{L^2(\mathbb{R}^3)})$$ So we have that $V$ is bounded with respect to the Laplacian and so Kato-Rellich theorem gives the thesis. Is there something worng in this reasoning?