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 Ft 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.
ASJC Scopus subject areas
- Applied Mathematics