A Weighted Estimate for Two Dimensional Schrödinger, Matrix Schrödinger and Wave Equations with Resonance of the First Kind at Zero Energy

We study the two dimensional Schrödinger operator, H = −∆ + V , in the weighted L(R) → L∞(R2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V (x)| . 〈x〉−4− and there is only s-wave resonance at zero of H, then ∥∥w−1(eitHPacf − 1 πit Ff )∥∥ ∞ ≤ C |t|(log |t|)2 ‖wf‖1, |t| > 2, with w(x) = log(2+ |x|). Here Ff = − 1 4 ψ〈ψ, f〉, where ψ is an s-wave resonance function. We also extend this result to wave and matrix Schrödinger equations with potentials under similar conditions.