In dimension $d\geq 3$ , we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schrödinger operators have neither resonances nor eigenvalues. If $d=2$ , we show that there are potentials with no resonances or eigenvalues away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on ${\mathbb R}^d$ . In odd dimensions $d\geq 3$ , we study the fundamental solution of the wave equation perturbed by such a potential. If the space variables are held fixed, it is superexponentially decaying in time