Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy

Abstract

We investigate L 1(ℝ4) → L ∞(ℝ4) dispersive estimates for the Schrödinger operator H = − Δ +V when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator F t satisfying ‖F t ‖ L 1→L ∞ < 1/log t for t > 2 such that We also show that the operator F t = 0 if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.