Schrödinger operators with complex-valued potentials and no resonances

Abstract

In dimension $d\geq 3$, we give examples of nontrivial, compactly supported, complex-valued potentials such that the associated Schrödinger operators have no resonances. If $d=2$, we show that there are potentials with no resonances away from the origin. These Schrödinger operators are isophasal and have the same scattering phase as the Laplacian on $\Real^d$. 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 super-exponentially decaying in time.