Even for self-adjoint Schr\"odinger operators on $\mathbb{R}^2$, the low energy behavior of the resolvent is rather complicated. In this talk we look at a new
approach to studying this problem which has the benefit of working for a large class of perturbations of $-\Delta$ on $\mathbb{R}^2$, including some
non-self-adjoint ones. We obtain asymptotic expansions of the resolvent near $0$ which have implications for quantities of interest in scattering theory.
This talk is based on joint work with Kiril Datchev.