Dispersive estimates for Schr\"odinger operators in dimension two with obstructions at zero energy

We investigate $L^1({\mathbb{R}}^2)\to L^{\mathrm{\infty }}({\mathbb{R}}^2)$ dispersive estimates for the Schr"odinger operator $H=-\mathrm{\Delta }+V$ 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 $|F_t|{L^1\to L^\infty} \lesssim 1$suchthat$$|e^{itH}P{ac}-F_t|_{L^1\to L^\infty} \lesssim |t|^{-1}, $\text{for}$ |t|>1.$$ 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.