Closure of partial differential operators on $L^2(\Omega)$ Let $\Omega:=\mathbb{R}^2\setminus\{0\}$. Consider the $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$
$$
H=-\partial_x^2-\partial_y^2+ \left(A_x\partial_x+A_y\partial_y\right)+(x^2+y^2)^{-1}
$$
with $A_i\in C^\infty(\Omega,\mathbb{C})$ given by 
$$
A(x,y):=\frac{(-y,x)}{x^2+y^2},
$$
defined over the dense domain $C_0^\infty(\Omega,\mathbb{C})$ of infinitely differentiable complex functions with compact support. There does exist a general way to derive its closure $\overline{H}$? 
Applying the definition i found that its domain is given by
$$
D(\overline{H})=\left\{u\in L^2(\Omega,\mathbb{C}),\ \exists v\in L^2(\Omega,\mathbb{C}),\ \int v\cdot\phi=-\int u \cdot H\phi\ \ \forall\phi\in C_0^\infty(\Omega,\mathbb{C})\right\}
$$
And the closure is then given by some weak form of $H$. But this form of the domain is quite unsatisfactory, what can I say about the existence and properties of the single weak derivatives $\mbox{w-}\partial^\alpha$? 
I hope I didn't make any mistake in my calculation
 A: So, let $D(H)=C_0^{\infty}(\Omega;\mathbb{C})$, and assume additionally that
$C_{\alpha}\in C(\overline{\Omega};\mathbb{C})$. Denote by $\overline{H}$
the minimal closed extension of $H$. The graph $\mathcal{G}(\overline{H})$
of $\overline{H}$ is a closure in $L^2(\Omega;\mathbb{C})\times 
L^2(\Omega;\mathbb{C})$ of the graph $\mathcal{G}(H)$ that consists of all
ordered pairs $\{u,Hu\}$ with $u$ ranging over $D(H)$. Consider a limit point
$\{u,v\}$ in $L^2(\Omega;\mathbb{C})\times L^2(\Omega;\mathbb{C})$ for a sequence
$\{u_k,Hu_k\}$ with $u_k\in D(H)$. Being convergent in $L^2(\Omega;\mathbb{C})$, 
sequences $\{u_k\}$ and $\{Hu_k\}$ are the Cauchy sequences. Hence by the a priori
estimate
$$
\|u_k-u_m\|_{H^2(\Omega;\mathbb{C})}\leqslant
C\bigl(\|Hu_k-Hu_m\|_{L^2(\Omega;\mathbb{C})}+\|u_k-u_m\|_{L^2(\Omega;\mathbb{C})}\bigr)
$$
follows that $v\in H_0^2(\Omega;\mathbb{C})$ defined as a closure of $D(H)=C_0^{\infty}(\Omega;\mathbb{C})$ 
in the Sobolev space $H^2(\Omega;\mathbb{C})$. Note that for a Lipschitz domain
$\Omega\subset\mathbb{R}^n$, the Sobolev space 
$$
H_0^2(\Omega;\mathbb{C})=\{u\in H^2(\Omega;\mathbb{C})\,\colon\,  
\partial^{\alpha}u|_{\partial\Omega}=0\,\forall\,\alpha\,\colon\,|\alpha|\leqslant 1\}.
$$
Thus $D(\overline{H})=H_0^2(\Omega;\mathbb{C})$ while 
$v=\overline{H}u=\sum_{|\alpha|\leqslant 2}\partial^{\alpha}u$. Unused condition 
$\sum_{|\alpha|\leqslant 2}\partial^{\alpha}C_{\alpha}=0$ is not needed here.
