I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. Here $u:\Omega \to \mathbb{R}^n$ is Lipschitz, and also I want to impose that all inputs to $L$ or $L^*$ are zero on $\partial \Omega$.
I would like to say that $L$ has the exact same eigenvalues as $L^*$, or at the very minimum, if $\lambda$ is an eigenvalue for $L$ then it is also an eigenvalue for $L^*$. Firstly I'm not sure if the "adjoint" operator $L^*$ is actually the adjoint of $L$ in the functional analysis sense. If it is, I am not totally sure what spaces I should be looking at. My first guess would be $L:W^{2,2}(\Omega) \to L^2(\Omega)$ but I don't really know.
Explanation and good reference books (specifically with treatment of adjoints of differential operators) are both appreciated.