Closure of partial differential operators on $L^2(\Omega)$

Question

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