Essential self-adjointness and uses for this operator?

Question

I was trying to come up with examples of self-adjoint partial differential operators and thought of this: Let $\Omega\subset\mathbb{R}^d$ be open and let $A:\Omega\to\mathbb{R}^d$ be a real $C^1(\Omega)$ vector field. Define the operator $(iA\cdot\nabla,H_0^1(\Omega))$, $$iA\cdot\nabla f(x):=\sum_{j=1}^d ia_j(x)\partial_{x_j}f(x),$$ where $a_j$ are the component functions of $A$. This is densley defined in $L^2(\Omega)$ and if $\text{div}(A)=0$, then this is also symmetric, as if $g\in H^1(\Omega)$, then \begin{equation}\begin{split}(ia_j\partial_{x_j}f|g) & =i\int_\Omega a_j\partial_{x_j}f\overline{g}\ dx \\ & =i\int_\Omega \partial_{x_j}f\overline{a_jg}\ dx \\ & =-i\int_\Omega f\overline{(a_j\partial_{x_j}g+g\partial_{x_j}a_j)}\ dx \\ & =\int_\Omega f\overline{i(a_j\partial_{x_j}g+g\partial_{x_j}a_j)}\ dx \end{split}\end{equation} and thus \begin{equation}\begin{split}(i A\cdot\nabla f|g) & =\sum_{j=1}^d(ia_j\partial_{x_j}f|g) \\ & =\sum_{j=1}^d\int_\Omega f\overline{i(a_j\partial_{x_j}g+g\partial_{x_j}a_j)}\ dx \\ & =\int_\Omega f\overline{\left(\sum_{j=1}^d ia_j\partial_{x_j}g+ig\sum_{j=1}^d\partial_{x_j}a_j\right)}\ dx \\ & = \int_\Omega f\overline{\left(\sum_{j=1}^d ia_j\partial_{x_j}g+ig\ \text{div}(A)\right)}\ dx \\ & =\int_\Omega f\overline{\sum_{j=1}^d ia_j\partial_{x_j}g}\ dx \\ & =\sum_{j=1}^d(f|i a_j\partial_{x_j} g) \\ & =(f|iA\cdot\nabla g). \end{split}\end{equation} My question then is in two parts: 1. Could this be essentially self-adjoint or somehow forced to be esentially sef-adjoint? 2. Do you know any places where this kind of operators might be used/of use?