We assume that $\Omega \subseteq \mathbb R^n$ is a bounded open set with smooth boundary and study the PDE $$ -\Delta u + \alpha u = f $$ with, say, zero Dirichlet boundary conditions along $\partial\Omega$. Here, $\alpha \in \mathbb C$.
I would like to understand the (weak) solution theory for different values of $\alpha$, in the context of Sobolev spaces.
We already know that the second-order differential operator has a positive discrete spectrum.
- When $\alpha \in \mathbb R$ and $\alpha > 0$, then we can apply Lax-Milgram.
- When $\alpha \in \mathbb R$ and $\alpha < 0$ is not in the negative of an eigenvalue, then the solution can be found in the literature as well.
But what happens, generally speaking, when $\alpha$ has a non-zero imaginary part? What are the essential techniques to prove the existence and uniqueness of the solution and its stability in the data?
The Poisson problem only serves as a model problem for more generally elliptic equations.