Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy

We investigate L¹(ℝ²) → L^∞(ℝ²) dispersive estimates for the Schrödinger operator H = -Δ+V when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the t^-1 decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy, then there is a time dependent operator F_t satisfying. We also establish a weighted dispersive estimate with t^-1 decay rate in the case when there is an eigenvalue at zero energy but no resonances.