In the quantum scattering theory one proves results of the following type. Let $H_0$ be the Laplacian $\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\frac{\partial^2}{\partial x_3^2}$ on $\mathbb{R}^3$. Let $V\colon \mathbb{R}^3\to \mathbb{R}$ be a potential satisfying some assumptions (e.g. decays fast enough at infinity). Then it is shown that the operator $H:=H_0+V$ has the same continuous spectrum as $H_0$ and there is 1-to-1 correspondence between their eigenfunctions (given by the Moller operators).
I am wondering if there are generalizations of this theory to the case when $H_0$ is the Laplacian plus a first order differential operator, and $V$ is another first order differential operator. I would also need the case when $H=H_0+V$ has no discrete spectrum.
