Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\quad u_{|\partial\Omega}=0$$

Now consider the elliptic operator $\Delta + b\cdot\nabla(-\Delta)^{-1}$ associated with, say, homogeneous Dirichlet boundary conditions, where $b$ is a smooth vector field (you can even take $b$ to be a constant vector field if that somehow makes this question answerable).

Is spectral analysis for such an operator possible? (i.e. existence of eigenvalues, properties of eigenfunctions, etc) I ask because such an operator (or rather a slightly more complicated version of it) appears as the linearization of a nonlinear operator that I'm studying. Any help/references would be greatly appreciated.