Time decay estimates for the wave equation with potential in dimension two

We study the wave equation with potential utt−Δu+Vu=0utt−Δu+Vu=0 in two spatial dimensions, with V   a real-valued, decaying potential. With H=−Δ+VH=−Δ+V, we study a variety of mapping estimates of the solution operators, cos(tH) and sin(tH)H under the assumption that zero is a regular point of the spectrum of H  . We prove a dispersive estimate with a time decay rate of |t|−12, a polynomially weighted dispersive estimate which attains a faster decay rate of |t|−1(log|t|)−2|t|−1(log|t|)−2 for |t|>2|t|>2. Finally, we prove dispersive estimates if zero is not a regular point of the spectrum of H.