Let $\Omega \in \mathbb{R}^n$ be a bounded domain with boundary of class $C^2$. Define \begin{cases} D(A) = H^2 \cap H_0^1(\Omega;\mathbb{C})\\ Au(x)=\Delta u(x)-V(x)u(x) \hspace{3mm} x \in \Omega \hspace{2mm}\mbox{a.e.} \end{cases}
where we assume that $V \in L^\infty(\Omega,\mathbb{R})$. I must show that $A$ is self-adjoint in $L^2(\Omega;\mathbb{C})$.
First I would like to check that $A$ is symmetric and my idea was integrating by parts and then see that $\rho (A) \cap \mathbb{R} \neq \emptyset$. But how to do deal with integration by parts when I have Laplacian?