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 = −∆+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 Ft satisfying ‖Ft‖L1→L∞ . 1 such that ‖ePac − Ft‖L1→L∞ . |t| −1, 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.