Generalized Laplacian?

I was wondering if any of you had ever encountered operators on $$L^2(\mathbb{R}^d)$$ of the form $$- \nabla \cdot A(x)\nabla$$ where $$A(x)$$ is some matrix field (viewed as $$L^2(\mathbb{R}^{d^2}$$)), and if so where can I find some literature about it ?

More precisely, I'm trying to get a lower bound, $$i.e$$ hopefully some function $$h(x)$$ such that $$- \nabla \cdot A(x)\nabla \geq h(x)$$ (in the sense of quadratic forms), when $$A(x) = (2|x| - |x|^{-1})xx^T$$.

Thanks a lot!

Indeed, such operators are often encountered in differential geometry, where a Laplace operator on a Riemannian manifold with metric $$g$$ is given by $$\Delta f = \frac{1}{\sqrt{\det g}} \nabla\cdot(g^{-1}\sqrt{\det g}\nabla f)$$ If matrix $$g$$ isn't positive definite, but it's non-degenerate, it's a slightly more complicated case of pseudo-Riemannian manifolds.