Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of $$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.

I also assume that $a(x) > 0$ for all $x$ and $a(x)$ is differentiable. However, I make no assumptions on the boundedness of $a$.

Does this operator have a core? If so, can it be identified explicitly?

Thanks.