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

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 {pipe}F _t{pipe}_L1→L∞ ≲ 1/log t for t > 2 such that (Formula presented.) 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.